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In: Current research topics in Galois geometryEditors: J. De Beule, L. Storme, pp. 141-167
ISBN 0000000000c© 2010 Nova Science Publishers, Inc.
Chapter 71
CODES OVER RINGS AND RING GEOMETRIES2
Thomas Honold∗†and Ivan Landjev‡§3
Abstract4
In this article, we bring together some recent results on special sets of points in5
coordinate projective geometries over finite chain rings. There is a clear coding theo-6
retic relevance of these results due to the strong connection between multisets of points7
in the chain ring geometries and so-called fat linear codes over finite chain rings. In8
Section 1, we introduce axiomatically projective and affine Hjelmslev spaces. An im-9
portant class of such spaces, obtained as coordinate geometries over finite chain rings,10
is given in Section 2. In Section 3, we define multisets of points in projective Hjelm-11
slev geometries and fat linear codes over finite chain rings. Furthermore, we state a12
result saying that these are essentially one and the same object. In Sections 4 and 5,13
we survey the known results on arcs and blocking sets in projective Hjelmslev planes.14
We include tables of the sizes of the largest known arcs in projective Hjelmslev planes15
over some small chain rings.16
Key Words: projective Hjelmslev geometry, projective Hjelmslev plane, finite chain ring,17
arcs, blocking sets, fat linear codes, Rédei type blocking sets, Witt vectors18
19
AMS Subject Classification: 51C05, 51E26, 51E21, 51E22, 94B05, 94B2720
1 Projective and affine Hjelmslev spaces21
?〈sec:hjelm〉?We start by introducing projective Hjelmslev spaces. The following axiomatic approach is22
due to Kreuzer [31–33, 35]. Let Π = (P ,L , I), I ⊆ P ×L , be an incidence structure. The23
∗Department of Information and Electronic Engineering, Zhejiang University, 38 Zheda Road, 310027Hangzhou, China. E-mail address: honold@zju.edu.cn
†Supported by the Open Project of Zhejiang Provincial Key Laboratory of Information Network Technologyand by the National Natural Science Foundation of China under Grant No. 60872063.
‡New Bulgarian University, 21 Montevideo str., 1618 Sofia, Bulgaria, and Institute of Mathematics andInformatics, BAS, 8 Acad. G. Bonchev str., 1113, Sofia, Bulgaria. E-mail address: ivan@math.bas.bg
§Supported by the Project Combined algorithmic and theoretical study of combinatorial structures betweenthe Research Foundation Flanders (Belgium) (FWO) and the Bulgarian Academy of Sciences, as well as by theStrategic Development Fund of the New Bulgarian University under Contract 357/14.05.2009.
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142 T. Honold and I. Landjev
sets P and L are referred to as sets of points and lines, respectively. A neighbour relation1
_ is defined on P and L by the following conditions:2
(N1) ∀x,y ∈ P : x _ y ⇐⇒∃S,T ∈ L ,S 6= T : {(x,S),(x,T ),(y,S),(y,T )} ⊆ I;3
(N2) ∀S,T ∈ L : S _ T ⇐⇒ for every point x with (x,S) ∈ I there is a point y with4
(y,T ) ∈ I and x _ y, and, conversely, for every y with (y,T ) ∈ I there is a point x with5
(x,S) ∈ I and y _ x.6
Given two points x,y with x 6 _ y we denote by x,y the unique line incident with both7
of them if such a line does exist. For a point x and a line S, we write x _ S if there exists a8
point y with (y,S) ∈ I, x _ y.9
Definition 1.1. An incidence structure Π = (P ,L , I) with neighbour relation _ is said to10
be a projective Hjelmslev space if it satisfies the following axioms:11
(H1) For any two points x,y ∈ P there exists a line S with (x,S) ∈ I, (y,S) ∈ I.12
(H2) Every line S ∈ L contains at least three points which are pairwise non-neighbours.13
(H3) Two lines S and T with S∩T 6= /0 are neighbours iff |S∩T | ≥ 2.14
(H4) For any x,y,z ∈ P , x _ y and y _ z imply x _ z.15
(H5) For any two lines S,T and any three points x,y,z with (x,S) ∈ I, (y,S) ∈ I, (x,T ) ∈ I,16
(z,T ) ∈ I, x 6_ y, x 6_ z, y _ z, we have S _ T .17
(H6) For a point x not incident with S ∈ L with x _ S, there always exist y,z ∈ P with18
y 6_ S, (z,S) ∈ I and (x,y,z) ∈ I.19
(H7) Let x∈ P , S∈L with x 6_ S and let y,z∈ S. For every (y′,x,y)∈ I and every (z′,x,z)∈20
I there exists a line T with (y′,T ) ∈ I, (z′,T ) ∈ I and S∩T 6= /0.21
The point set T ⊆ P is called a Hjelmslev subspace of Π if for every two distinct points22
x,y ∈ P , there exists a line L ∈ L(T ) = {L ∈ L | L ⊆ T } with (x,L) ∈ I, (y,L) ∈ I. We23
write x _ T if there exists a point y ∈ T with x _ y. Every Hjelmslev subspace T forms a24
projective Hjelmslev space (T ,L(T ), IT ) of its own, where IT = I∩(T ×L(T )). For every25
X ⊆ P we define the hull 〈X 〉 as the intersection of all Hjelmslev subspaces containing X .26
The set X ⊆ P is said to be independent if for any x ∈ X we have x 6_ 〈X \{x}〉.27
Definition 1.2. The point set B is a basis of Π if 〈B〉= P and B is independent.28
The dimension of a projective Hjelmslev space Π is defined as dimΠ = |B|−1. In what29
follows, we consider only finite-dimensional Hjelmslev spaces.30
2 Coordinate Hjelmslev geometries31
?〈sec:galois〉?An important class of projective Hjelmslev spaces can be obtained as coordinate geometries32
from modules over so-called finite chain rings. We review only the most basic properties of33
this class of finite rings, and refer the reader for a detailed treatment to [7, 39, 40, 42].34
An associative ring R with identity (1 6= 0) is called a left (right) chain ring if the lattice35
of left (resp., right) ideals of R forms a chain. In the finite case, |R| < ∞, this condition is36
left-right symmetric and equivalent to R being a local principal ideal ring. In what follows37
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Codes over rings and ring geometries 143
the Jacobson radical rad(R) of R (which we assume to be a finite chain ring from now1
on) will be denoted by N, so that R/N ∼= Fq is a finite field and N = Rθ = θR for any2
θ ∈ N \N2. Furthermore, there exists an integer m ≥ 1 (called the length or nilpotency3
index of R) such that Nm−1 6= {0}, Nm = {0}, and every left or right ideal of R belongs4
to the chain R > N > N2 > · · · > Nm−1 > {0} of two-sided ideals Ni = Rθi = θiR. The5
finite chain rings of length m = 1 are just the fields Fq and thus trivial from our point of6
view. For the smallest non-trivial case m = 2, a detailed description and classification of the7
corresponding rings will be given in Section 4.8
Let MR be a finite free (right) module over R of rank rkM ≥ 3. Denote by P and L9
the set of all free rank 1, respectively free rank 2, submodules of MR and by I ⊆ P ×L set-10
theoretical inclusion. The incidence structure (P ,L , I) satisfies (H1)–(H7) and, therefore, is11
a projective Hjelmslev space. If rkM = k, this incidence structure is referred to as the (right)12
(k−1)-dimensional projective Hjelmslev geometry over the chain ring R and is denoted by13
PHG(RkR). (Since MR ∼= Rk
R, this is no essential restriction.)14
Let R be a chain ring with |R| = qm, R/N ∼= Fq. We consider the projective Hjelmslev15
space Π = (P ,L , I) = PHG(RkR). Two points x = xR and y = yR are called i-neighbours,16
i = 0,1, . . . ,m, if |x∩ y| ≥ qi. This fact is denoted by x _ iy. Two lines S and T are i-17
neighbours if for every point x on S there exists a point y on T with x _ iy, and conversely,18
for every y on T there exists x on S with y _ ix. Every two points (lines) are 0-neighbours;19
1-neighbourhood is the same as the neighbour relation defined by (N1) and (N2).20
For every i ∈ {0,1, . . . ,m}, the relation _ i is an equivalence relation on P , as well21
as on L . The equivalence classes of this relation are denoted by [x](i), x ∈ P , respectively22
[S](i), S ∈ L . The set of all equivalence classes of _ i on points, resp. lines, is denoted23
by P (i), resp. L(i). We denote by π(i) the natural homomorphism π(i) : R → R/Rθi, where24
Rθ = radR. By π(i), we denote the mapping induced by π(i) on the Hjelmslev subspaces of25
Π.26
Below we state some facts on the combinatorics and the structure of the projective27
Hjelmslev geometries PHG(RkR) (cf. [2, 10, 12, 19, 29–31, 33, 44]).28
?〈fact:thm1〉?Fact 2.1. Let Π = (P ,L , I) = PHG(RkR), where R is a chain ring with |R|= qm, and R/N ∼=29
Fq. For every two integers s, t with 0 ≤ t ≤ s ≤ k, the number of all (s− 1)-dimensional30
Hjelmslev subspaces through a fixed (t−1)-dimensional subspace is equal to31
q(s−t)(k−s)(m−1)[
k− ts− t
]q,32
where33 [ks
]q=
(qk−1)(qk−1−1) · · ·(qk−s+1−1)(qs−1)(qs−1−1) · · ·(q−1)
.34
Moreover, the number of points that are i-th neighbours to a fixed point is q(k−1)(m−i) for all35
i = 1, . . . ,m.36
The next few results explain the structure of the geometries PHG(RkR) in some more37
detail.38
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144 T. Honold and I. Landjev
Define a new incidence relation J(i) ⊆ P (i)×L(i) by1
([x](i), [S](i)) ∈ J(i) ⇔∃x′ ∈ [x](i),∃S′ ∈ [S](i) : (x′,S′) ∈ I.2
?〈fact:thm0〉?Fact 2.2. The incidence structure (P (i),L(i),J(i)) is isomorphic to the projective Hjelmslev3
geometry PHG((R/Rθi)kR/Rθi). In particular, (P (1),L(1),J(1)) is isomorphic to PG(k−1,q).4
Let ∆1,∆2 be two Hjelmslev subspaces with dim∆1 ≤ dim∆2. We write ∆1 _ i∆2 if5
π(i)(∆1) ⊆ π
(i)(∆2). Note that under this definition _ i is not symmetric. We consider6
again Π = (P ,L , I) = PHG(RkR), where R is a chain ring with |R|= qm, R/N ∼= Fq. Let us7
fix a Hjelmslev subspace Σ with dimΣ = u−1 and an integer i, 1 ≤ i ≤ m−1. Denote by8
Pi(Σ) the set of all points x with x _ iΣ. Now set9
P ={
∆∩ [x]m−i | x ∈ Pi(Σ),dim∆ = u−1, ∆ _ iΣ, ∆∩ [x]m−i 6= /0
}. (1)10
It can be proved that the sets ∆∩ [x]m−i are either disjoint or coincide for the various choices11
of ∆.12
Let S ∈ L . We say that the “point” x = ∆∩ [x]m−i ∈P is incident with the line S if13
∆∩ [x]m−i∩S 6= /0.14
This defines an incidence relation I′ ⊆ P×L . For two lines S and T we write S ∼ T if S15
and T are incident with the same points of P. Clearly ∼ is an equivalence relation on L .16
Denote by L a set of representatives from the different equivalence classes of lines under17
∼, which have nonempty intersection with at least one of the sets ∆∩ [x]m−i. Let J be the18
incidence relation induced by I′ on P×L. With the above notation, we have the following19
result.20
〈fact:thm3〉Fact 2.3. The incidence structure (P,L,J) can be embedded isomorphically into21
PHG((R/Rθm−i)kR/Rθm−i). The missing part consists of the points of a (k − u − 1)-22
dimensional Hjelmslev subspace.23
The (k− 1)-dimensional affine Hjelmslev geometry AHG(Rk−1R ) is defined as the in-24
cidence structure obtained from PHG(RkR) by deleting a neighbour class of hyperplanes.25
Equivalently, it can be defined as the incidence structure having as points all (k−1) tuples26
(α1, . . . ,αk−1), αi ∈ R, and as lines all cosets of free rank 1 submodules of Rk−1R . If in the27
discussion preceding Fact 2.3, we take Σ to be a point, say x, then P = [x]i and we get the28
following result.29
?〈fact:thm2〉?Fact 2.4. ([x]i,L,J)∼= AHG((R/Rθ
m−i)k−1R/Rθm−i
).30
In particular,31
([x]m−1,L)∼= AG(k−1,q).32
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Codes over rings and ring geometries 145
3 Multisets of points in projective Hjelmslev geometries and lin-1
ear codes over finite chain rings2
?〈sec:codmul〉?3.1 Multisets of points in PHG(RkR)3
?〈ssec:multisets〉?Let Π = PHG(RkR) = (P ,L , I) be a finite-dimensional projective Hjelmslev geometry over4
the chain ring R.5
Definition 3.1. A multiset in Π is a mapping K : P → N0.6
The mapping K is extended to the subsets of P by7
K(Q ) = ∑x∈Q
K(x) for Q ⊆ P . (2)8
The integer K(x) is called the multiplicity of the point x. The integer K(P ) = ∑x∈P K(x)9
is called the cardinality or length of the multiset K and is denoted by |K|. The support10
suppK of K is defined by suppK = {x ∈ P |K(x) > 0}. For a multiset K in Π we define its11
hull 〈K〉 ≤ RkR by12
〈K〉= ∑xR∈suppK
xR. (3)13
Clearly, 〈K〉 can be considered as the set of all points x = xR with x ≤ 〈K〉.14
Given a set of points Q ⊆ P , we define the characteristic multiset χQ by15
χQ (x) ={
1 if x ∈ Q0 otherwise.
16
All multisets K satisfying K(x) ∈ {0,1} for every x ∈ P arise in this manner from their17
supports. Such multisets are said to be projective and may be tacitly identified with their18
supports.19
The multiset K induces in a natural way multisets K(i) in π(i)(Π) by20
K(i) : P (i) → N0 : [x]i 7→ K([x]i)21
for i = 0,1, . . . ,m. Note that π(i)(〈K〉) = 〈K(i)〉.22
〈l:rank〉Definition 3.2. Denote by κi the rank of the R-module 〈K(i)〉.23
In geometric language, κi − 1 is the dimension of the smallest Hjelmslev subspace of24
π(i)(Π) containing all points of suppK(i).25
?〈dfn:mset-equiv〉?Definition 3.3. Two multisets K in Π and K′ in Π′ are said to be equivalent if there exists26
a bijective R-semilinear mapping ψ : 〈K〉R → 〈K′〉R such that K(x) = K′(ψ(x))
for every27
point x ∈ 〈K〉.28
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146 T. Honold and I. Landjev
3.2 Linear codes over finite chain rings1
?〈ssec:codes〉?Let R be a chain ring with |R| = qm, R/N ∼= Fq, let θ be a generator of N, and consider2
the set Rn of all n-tuples over R. The set Rn has the structure of an (R-R)-bimodule with3
respect to component-wise addition and left/right multiplication by elements from R. We4
say that θi is the period of the vector x ∈ Rn if i is the smallest non-negative integer with5
θix = 0 (equivalently, with x ∈ Rnθm−i). We denote this by θm−i ‖ x. The set of vectors in6
Rn of period θm is denoted by (Rn)∗. Since Rθi = θiR for all i ≥ 0, the concept of period is7
left-right symmetric even for non-commutative chain rings.8
?〈dfn:code〉?Definition 3.4. A code C of length n over R is a non-empty subset of Rn. The vectors of C9
are called codewords. The code C is left (resp., right) linear if it is an R-submodule of RRn10
(resp., of RnR). A linear code is one which is either left or right linear.11
Definitions and results in the sequel will be stated for left linear codes, most of them12
having obvious right counterparts.13
A partition λ ` n of an integer n is a sequence of non-negative integers λ0 ≥ λ1 ≥14
λ2 ≥ . . . with ∑i≥0 λi = n. The trailing zeros of this sequence will be suppressed. The15
following theorem generalizes the structure theorem for finite abelian p-groups (see e.g. [38,16
Ch. 15,§ 2]):17
?〈thm:cyclic〉?Theorem 3.5 ( [21]). Every linear code C over a chain ring R is a direct sum of cyclic18
R-modules. The partition λ = (λ1, . . . ,λk) ` logq|C | satisfying19
RC ∼= R/Nλ1 ⊕·· ·⊕R/Nλk (4)20
is uniquely determined by RC . Moreover, the partition µ = λ′ ` logq|C | conjugate to λ has21
components µi = dimR/N(θi−1C/θiC ).22
?〈dfn:shape〉?Definition 3.6. The shape of a linear code C over R is the partition
λ = (λ1, . . . ,λk) ` logq|C |,
which satisfies RC ∼= R/Nλ1 ⊕ ·· · ⊕R/Nλk . The partition λ′ conjugate to λ is called the23
conjugate shape of C . The integer k = λ′1 = dimR/N(C/θC ) is called the rank of C and is24
denoted by rkC . A subset {x1, . . . ,xk} ⊆ C \{0} is called a basis of C if RC = Rx1⊕·· ·⊕25
Rxk.26
?〈dfn:gmatrix〉?Definition 3.7. Let C ≤ RRn be a linear code of rank rkC = k. A generator matrix of C is27
a k×n-matrix having as its rows a basis of C , so that, in particular, C = {xG;x ∈ Rk}.28
For two vectors u = (u1, . . . ,un) ∈ Rn and v = (v1, . . . ,vn) ∈ Rn we define their inner29
product u ·v by30
u ·v := u1v1 + · · ·+unvn. (5)31
Given a code C ⊆ Rn, we define32
C⊥ = {y ∈ Rn | x ·y = 0 for every x ∈ C},33
⊥C = {y ∈ Rn | y ·x = 0 for every x ∈ C}.34
The linear code C⊥ ≤ RnR (resp., ⊥C ≤ RRn) is called the right (resp., left) orthogonal code35
of C .36
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Codes over rings and ring geometries 147
〈thm:perp〉Theorem 3.8 ( [21]). Let C ≤ RRn be a linear code of shape λ = (λ1, . . . ,λn) and rank1
λ′1 = k.2
〈l:perp-shape〉1. The orthogonal code C⊥ has shape (m− λn,m− λn−1, . . . ,m− λ1) and conjugate3
shape (n−λ′m,n−λ′m−1, . . . ,n−λ′1). In particular, C is free as an R-module if and4
only if C⊥ is free if and only if rk(C⊥) = n− k;5
〈l:perp-perp〉2. ⊥(C⊥) = C ;6
?〈l:perp-lattice〉?3. if in addition C ′ ≤ RRn then (C ∩C ′)⊥ = C⊥+C ′⊥, and (C +C ′)⊥ = C⊥∩C ′⊥.7
?〈cor:row-column〉?Corollary 3.9. Let G ∈ Mm,n(R) be any matrix. The linear codes C ≤ RRn and D ≤ RmR8
generated by the rows and columns of G, respectively, have the same shape.9
?〈dfn:kmatrix〉?Definition 3.10. A parity check matrix of a linear code C ≤ RRn is an (n−λ′m)×n-matrix10
whose rows form a basis of the orthogonal code C⊥11
Note that if H is a parity-check matrix of C , then by Part 2 of Theorem 3.8 we have12
x ∈ C if and only if x ·HT = 0. The number of (and periods of the) columns of H are13
determined by Part 1 of Theorem 3.8.14
For x = (x1, . . . ,xn) ∈ Rn we set15
ai(x) = |{ j | 1 ≤ j ≤ n and θi ‖ x j}|.16
?〈dfn:type〉?Definition 3.11. The sequence(a0(x), . . . ,am(x)
)is called the type of the word x ∈ Rn.17
?〈dfn:isom〉?Definition 3.12. An automorphism of the code Rn is a bijective mapping φ : Rn → Rn which18
satisfies ai(x−y) = ai(φ(x)−φ(y)
)for all x,y ∈ Rn and all i ∈ {0,1, . . . ,m}.19
?〈dfn:cisom〉?Definition 3.13. Two codes C1,C2 ⊆ Rn are said to be isomorphic (resp., semilinearly iso-20
morphic) if there exists a code automorphism (resp., semilinear code automorphism) φ of21
Rn with φ(C1) = C2.22
3.3 Equivalence of multisets of points and linear codes23
?〈ssec:equiv〉?Definition 3.14. A linear code C ≤ RRn is said to be fat if for every i ∈ {1, . . . ,n} there24
exists a codeword c = (c1,c2, . . . ,cn) ∈ C with ci ∈ R× (i.e. ci is a unit in R).25
Let C ≤ RRn be a fat linear code. Let S = (c1, . . . ,ck) be a sequence of (not necessarily26
independent) generators for RC and let G∈Mk,n(R) be the k×n-matrix with rows c1, . . . ,ck.27
Denote the columns of G by g1, . . . ,gn. Since C is fat and c1, . . . ,ck generate C , the vectors28
g j have period θm and thus define points g jR in the projective (right) Hjelmslev geometry29
(P ,L ,I ) = PHG(RkR). We define the multiset KS induced by the generating sequence S of30
C as31
KS : P → N0 : x 7→ |{ j | x = g jR}|. (6)32
We say that the multiset KS and the code C = Rc1 + · · ·+ Rck are associated. By the33
definition of KS, we have |KS| = n. Furthermore, the modules 〈KS〉 and RC have the same34
shape and, in particular, the same cardinality; see [21].35
The following theorem is a generalization of a similar result by Dodunekov and Simonis36
[11] about linear codes over finite fields.37
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148 T. Honold and I. Landjev
?〈thm:codmul〉?Theorem 3.15. For every multiset K of length n in PHG(RkR) there exists a fat linear code1
C ≤ RRn and a generating sequence S = (c1, · · · ,ck) of C which induces K. Two multisets K12
in PHG(Rk1R ) and K2 in PHG(Rk2
R ) associated with fat (left) linear codes C1 and C2 over R,3
respectively, are equivalent if and only if the codes C1 and C2 are semilinearly isomorphic.4
?〈dfn:ktype〉?Definition 3.16. Let K : P → N0 be a multiset in Π = PHG(RkR). A hyperplane ∆ in Π is5
said to have the K-type (a0,a1, . . . ,am), where6
ai = ∑x _ i∆,x 66_ i+1∆
K(x) for0 ≤ i ≤ m.7
By duality (cf. Theorem 3.8), every hyperplane ∆ in PHG(RkR) can be considered as a
set of points, whose homogeneous coordinates (x1, . . . ,xk) satisfy a linear equation
r1x1 + r2x2 + . . .+ rkxk = 0,
where at least one of the ri’s is a unit in R. Let C be a fat linear code associated with K,8
and let GS be a k× n-matrix whose sequence S of rows generates C and satisfies KS =9
K. All codewords of C which belong to the cyclic submodule R(r1, . . . ,rk)GS ≤ RC are10
called codewords associated with the hyperplane ∆ (relative to the choice of the generating11
sequence S). There is a connection between the K-type of a hyperplane in Π and the number12
of codewords of a given type in C associated with that hyperplane.13
?〈thm:nowords〉?Theorem 3.17. Let K be a multiset in PHG(RkR) and let C be a fat linear code over R14
associated with K. For each hyperplane ∆ of K-type (0, . . . ,0,a j,a j+1, . . . ,am), with a j 6= 0,15
0 ≤ j ≤ m, there exist exactly qm−s−qm−s−1 codewords in C of type16
(0, . . . ,0︸ ︷︷ ︸s
,a j, . . . ,am+ j−s−1,m
∑i=m+ j−s
ai) ( j ≤ s ≤ m−1) (7)17
which are associated with ∆.18
For a multiset K in PHG(RkR), the numbers κi = rk〈K(i)〉 (Definition 3.2) determine the19
shape of every fat linear code C ≤ RRn associated with K.20
〈thm:card〉Theorem 3.18. Let K be a multiset in PHG(RkR) associated to the fat linear code C . Then
C has conjugate shape λ′ = (κm,κm−1, . . . ,κ1), and, in particular,
|C |= qκ1+κ2+···+κm .
3.4 Some classes of codes defined geometrically21
?〈ssec:geocodes〉?Consider the Hjelmslev geometry Π = (P ,L , I) = PHG(RkR). The linear code C associated22
with the multiset K defined by K(x) = 1 for all x ∈ P , is called the k-dimensional simplex23
code over R and is denoted by Sim(k,R). The code Sim(k,R) has length q(k−1)(m−1)[k
1
]q,24
and by Theorem 3.18 it has shape mk (i.e. its shape consists of k parts equal to m), in25
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Codes over rings and ring geometries 149
particular |Sim(k,R)|= qkm. All hyperplanes ∆ in Π have the same K-type (a0,a1, . . . ,am),1
where2
a0 = q(k−1)(m−1)
([k1
]q−[
k−11
]q
)= q(k−1)m,
a j = q(k−2)(m−1)[
k−11
]q
(qm− j−qm− j−1) , j = 1, . . . ,m−1,
am = q(k−2)(m−1)[
k−11
]q.
The dual code Sim(k,R)⊥ is called the k-th order Hamming code over R and is de-3
noted by Ham(k,R). It is free of rank q(k−1)(m−1)[k
1
]q − k, in particular |Ham(k,R)| =4
qmq(k−1)(m−1)[k1]q−mk. A parity check matrix and a generator matrix for Ham(k,R) may be5
obtained similarly to the special case of R being a field.6
4 Arcs in projective Hjelmslev planes7
〈sec:arcs〉4.1 The maximal arc problem8
?〈ssec:maxarcpbm〉?〈dfn:arcs〉Definition 4.1. A multiset K in (P ,L , I) is called a (k,n)-arc if9
(i) K(P ) = k.10
(ii) K(L)≤ n for every line L ∈ L .11
According to this definition, a (k,n)-arc is also a (k,n′)-arc for every integer n′ ≥ n. For12
this reason we shall usually assume that n is chosen to be minimal, i.e. there exists at least13
one line L0 ∈ L with K(L0) = n (but there are exceptions). Moreover, sometimes we say14
“n-arc” in place of “(k,n)-arc” without referring to the cardinality of K.15
Of course, Definition 4.1 also makes sense for other incidence structures. In the clas-16
sical cases of PG(2,q) or AG(2,q) (which can be considered as special cases of projective17
Hjelmslev planes) a lot of research has been done on arcs and many results are known. For18
an overview, see, for example [9]. Some of these results will be used in the sequel.19
The arcs considered in this section will be projective and can be identified with sets of20
points, as described earlier.21
Furthermore, for the rest of this survey, we will confine ourselves to the case of finite22
chain rings R of length 2, i.e. the case |R| = q2, R/N ∼= Fq. The classification of all those23
rings is known and summarized in the following result.24
?〈fact:classify〉?Fact 4.2 ( [43, Th. 4] or [8, Th. 6]). Suppose R is a finite chain ring with |R|= q2, R/N ∼= Fq,25
where q = pr. Then26
(i) either R has characteristic p2 and is isomorphic to the Galois ring GR(q2, p2) of order27
q2 and characteristic p2, defined as GR(q2, p2) = Zp2 [X ]/(h) where h ∈ Zp2 [X ] is a28
(monic) polynomial of degree r which is irreducible modulo p, or29
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150 T. Honold and I. Landjev
(ii) R has characteristic p and for some σ ∈ Aut(Fq) is isomorphic to the ring1
Fq[X ;σ]/(X2) of σ-dual numbers over Fq, defined as the set of all a0 + a1X ∈ Fq[X ]2
with operations (a0 + a1X) + (b0 + b1X) := a0 + b0 + (a1 + b1)X, (a0 + a1X)(b0 +3
b1X) := a0b0 +(a0b1 +a1σ(b0)
)X.4
Moreover, the r +1 different rings listed in (i), (ii) are pairwise non-isomorphic.5
In the sequel, we will also refer to these rings as Gq := GR(q2, p2), Sq := Fq[X ]/(X2),6
and T(i)q := Fq[X ;σi]/(X2) for 1 ≤ i ≤ r−1, where σ denotes the Frobenius automorphism7
of Fq. Furthermore we will use the abbreviations Tq = T(1)q and T◦
q = T(r−1)q .18
Denote by mn(R3R) the maximal value of k for which there exists a (k,n)-arc in9
PHG(R3R). The problem of determining the exact values of mn(R3
R) for various values of10
n and for various rings R is central and has a clear coding theoretic relevance.11
4.2 A general upper bound on the size of an arc12
?〈ssec:upperbound〉?The following theorems provide upper bounds on the size of a (k,n)-arc in PHG(R3R) [37].13
?〈thm:bound〉?Theorem 4.3. Let K be a (k,n)-arc in PHG(R3R) where |R| = q2, R/N ∼= Fq. Suppose14
there exists a neighbour class of points [x] with K([x]) = u and let ui, i = 0,1, . . . ,q, be the15
maximum number of points on a line from the i-th parallel class in the affine plane defined16
on [x]. Then17
k ≤ q(q+1)n−qq
∑i=0
ui +u.18
Proof. Let {Li | i = 0,1, . . . ,q} be a set of q + 1 lines no two of which are neighbours andsuch that K([x]∩Li) = ui. For every i ∈ {0, . . . ,q}, denote by L( j)
i , j = 1, . . . ,q, the q linesin PHG(R3
R) that coincide with Li on [x]. The sum of the multiplicities of the points fromL( j)
i not in [x] does not exceed n−ui, which gives the estimate
k = K([x])+q
∑i=0
q
∑j=1
K(L( j)i \ ([x]∩Li))
≤ u+q
∑i=0
q
∑j=1
(n−ui)
= u+q
∑i=0
q(n−ui)
= u+q(q+1)n−qq
∑i=0
ui.
19
1The latter reflects the fact that T(r−1)q is isomorphic to the opposite ring of T(1)
q . Note that the smallest case
where a symbol T(i)q cannot be avoided is q2 = 256.
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Typically, the numbers ui are unknown. We can use some simple estimates to get a1
more convenient form for the above upper bound. From the obvious inequality ui ≥ du/qe,2
we get3
k ≤ q(q+1)(n−du/qe)+u. (8)4
Fix a point y ∈ [x] and let S0, . . . ,Sq be lines through y, no two of which are neighbours.Without loss of generality, we assume that Li _ Si for i = 0, . . . ,q. Set si = K([x]∩ Si)−K(y). Clearly, K(y)+ si ≤ ui. Then
k ≤ q(q+1)n−qq
∑i=0
ui +u
≤ q(q+1)n−qq
∑i=0
(K(y)+ si)+u
= q(q+1)n−q(q+1)K(y)−q(u−K(y))+u
= q2(n−K(y))+q(n−u)+u.
Since we may certainly assume that K(y)≥ 1, the last inequality simplifies to5
k ≤ q2(n−1)+q(n−u)+u.6
〈thm:genbound〉Theorem 4.4.
mn(R3)≤ max1≤u≤min{µn(q),q2}
min{u(q2 +q+1),
q2(n−1)+q(n−u)+u,q(q+1)(n−du/qe)+u},
where µn(q) denotes the maximal size of a (k,n)-arc in AG(2,q).7
For small values of n, we can derive somewhat better bounds.8
?〈thm:oval_bound〉?Theorem 4.5. m2(R3)≤
{q2 +q+1 for q even,q2 for q odd.
(9)9
In case of equality, we have10
(i) for q even, K([x]) = 1 for every [x] ∈ P (1);11
(ii) for q odd, K([x]) ≤ 1 for every [x] ∈ P (1). Moreover, the neighbour classes with12
K([x]) = 0 form a line in the factor plane (P (1),L(1),J (1))∼= PG(2,q).13
?〈thm:q=9〉?Theorem 4.6. m3(R3R)≤ 2q2−q+3, for every q ≥ 5.14
Note that in the cases q = 2,3, the exact value of m3(R3R) is known. It is 10 for the rings15
of cardinality 4, 19 for R = Z9, and 18 for R = F3[X ]/(X2). For q = 4, we have the bounds16
29 ≤ m3(R3)≤ 30 for all three rings G4, S4, T4.17
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152 T. Honold and I. Landjev
4.3 Constructions for arcs1
?〈ssec:constructions〉?In this section, we present general constructions for arcs in projective Hjelmslev planes.2
Throughout this section, R will be a chain ring with |R|= q2, R/N ∼= Fq, and Π = (P ,L , I)3
will be the projective Hjelmslev plane PHG(R3R).4
〈ex:ex1〉Example 4.7. For values of n close to q2 + q, the exact value of mn(R3R) can be easily5
computed. For every chain ring R, with |R|= q2, R/N ∼= Fq, and every integer s = 0,1, . . . ,q6
mq2+s(R3R) = q4 +q2s+qs.7
Denote by F the point set obtained in the following way. Fix a line L. Take in F the8
points of the line L plus q− s− 1 additional line segments parallel to L∩ [xi] in each of9
the neighbour classes [xi] incident with [L] in the factor geometry. The multiset χP −χF is10
easily checked to be the desired arc. The upper bound is obtained from Theorem 4.4.11
Example 4.8. Now we describe a general construction for (q4− q2− 2q + 1,q2− 1)-arcs12
in PHG(R3R) that does not depend on the underlying ring. Remarkably, this construction13
is better than the “triangle construction” which yields a (q4−2q2 +1,q2−1)-arc χP −χF14
as the complement of a “triangle” F (F consists of a neighbour class of lines and two15
additional lines that are not neighbours).16
Fix a point class [x0] and a line class [L0] incident with [x0] in the factor plane. Set17
[L0] = {[xi] | i = 0, . . . ,q}.18
Furthermore, denote by [Li], 1≤ i≤ q, the other line classes through [x0] in the factor plane.19
Consider the set K containing the following points:20
1) The complement of a (2q−1)-blocking set in the affine plane induced on [x0] (which is21
isomorphic to AG(2,q)). Thus K contains (q−1)2 points from [x0].22
2) The line segments from the point classes [x1], . . . , [xq] together with q additional lines23
(containing the segments in [xi], i = 1, . . . ,q) form a structure isomorphic to AG(2,q).24
In every class [xi], choose q−2 line segments (having the direction of [L0]) such that the25
resulting q(q−2) line segments form the complement of a blocking set in AG(2,q).26
3) From each of the remaining point classes [y], select the following points. If [y] ∈ [Li],27
take the q2−q points from q−1 parallel line segments having the direction of the line28
[yxi].29
The total number of points is (q− 1)2 + q · q(q− 2) + q2(q2 − q) = q4 − q2 − 2q + 1.30
A line in [L0] meets [x0]∩K in at most q− 1 points and at most q− 1 of the sets [xi]∩K,31
i = 1, . . . ,q, in q points, i.e., it contains at most q−1+(q−1)q = q2−1 points from K.32
A line in the class [yx0], y 6_ L0, meets [x0]∩K in at most q−1 points and each of the33
other q sets [z]∩K in exactly q−1 points. Hence, such a line contains at most q−1+q(q−34
1) = q2−1 points from K.35
Finally, a line in the class [yxi], y 6_ L0, i 6= 0, meets one set [z]∩K in at most q points,36
q−1 such sets in q−1 points and one set (the set [xi]∩K) in q−2 points. Therefore, such37
a line contains at most q+(q−1)2 +q−2 = q2−1 points.38
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Thus the arc defined above has the desired parameters.1
This construction can be further improved if we take the blocking set on [L0] to consist2
of two lines that meet on [x0]. Furthermore, we replace the q−1 points from [x0] that form3
a line segment in a direction different from that of L0 by q− 2 collinear points in [x1] that4
again have a direction different from that of L0 and are not already part of the blocking5
set on [L0]. It is an easy check that the size of the arc is increased by 1 and we get a6
((q3 +q2−2)(q−1),q+1)-arc.7
For q2 = 9,16,25, this construction gives: m8(R3R)≥ 68, for q2 = 9,m8(R3
R)≥ 234, for8
q2 = 16, and m8(R3R)≥ 592, for q2 = 25.9
The exact formula for mn(R3R) in the range q2 ≤ n ≤ q2 + q presented in Example 4.710
may also be written as mn(R3R) = q4 + q3 + q2 − (q2 + q− n)(q2 + q). From this point of11
view it says that the complementary((q2 + q− n)(q2 + q),q2 + q− n
)-blocking set has12
the same cardinality as the (generally non-projective) sum of q2 + q− n lines. It seems13
reasonable to conjecture that the lower bound mn(R3R)≥ q4 +q3 +q2− (q2 +q−n)(q2 +q)14
holds for all n. (For small values of n, this lower bound is even rather weak.) The following15
theorem extends the range of integers n, for which the lower bound is known to hold, to16
q2−bq/2c ≤ n ≤ q2 +q.17
?〈thm:triangle〉?Theorem 4.9. For every chain ring R with |R| = q2, R/ radR ∼= Fq, and every integer s =18
1,2, . . . ,bq/2c, the following inequality holds:19
mq2−s(R3R)≥ q4−q2s−qs. (10)20
Proof. We will prove the existence of a(t(q2 +q), t
)-blocking set in PHG(R3
R) for q+1 ≤21
t ≤ b3q/2c except in the case (q, t) = (3,4), which is covered by the subsequent Exam-22
ple 4.10.23
Choose point classes [x0], [x1], [x2] and line classes [L0], [L1], [L2] which form a triangle24
in the factor plane PG(2,q), indexed in such a way that [xi] is incident with [Li−1] and [Li]25
(indices taken modulo 3). There exist (unique) integers t1, t2, t3 satisfying t = t1 + t2 + t326
and 1 ≤ t1 ≤ t2 ≤ t3 ≤ t1 + 1. In each point class [x] incident with [Li] but different from27
the vertices [xi] and [xi+1], choose ti parallel line segments in the direction of [Li]. In each28
class [xi] choose ti−1 + ti parallel line segments in the direction of [Li]. This is possible,29
since ti−1 + ti ≤ t − t1 = t −bt/3c = d2t/3e ≤ q. It is clear that the resulting point set in30
PHG(R3R) blocks every line outside [L1]∪ [L2]∪ [L3] exactly t times. 2 Every line L ∈ [Li] is31
blocked ti +ti+1 times by the line segments in [xi+1]. Since t3 = dt/3e ≤ dq/2e< q, we have32
q+ ti + ti+1 > t. In order to have K(L)≥ t, it is therefore enough to ensure that L is blocked33
at least once by the line segments chosen in [Li]\ [xi+1]. The q2 line segments in [Li]\ [xi+1]34
(as points) together with the q2 lines in [Li] and the q point classes [y1] = [xi], [y2], . . . , [yq] (as35
lines) form an incidence structure isomorphic to AG(2,q). Our task is to arrange the ti−1 +ti36
line segments in [y1] and the ti line segments in each class [y j], 2 ≤ j ≤ q, in such a way37
that they form a blocking set in AG(2,q).3 Since ti ≥ 1, we may assume that q of these line38
2The construction so far can also be seen as taking the sum of t = t1 + t2 + t3 lines in PHG(R3R), where
ti lines are chosen from [Li] in such a way that they have a line segment in [xi+1] in common. To make theresulting multiset projective, the ti-fold line segment in [xi+1] is replaced by ti line segments in [xi+1] havingdirection [Li+1] and not already chosen during the first step.
3Note that the special lines [y1], [y2], . . . , [yq] are blocked by construction, since ti−1 + ti > ti ≥ 1.
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154 T. Honold and I. Landjev
segments, one from each class [y j], are collinear. The remaining ti−1 +ti−1+(q−1)(ti−1)1
line segments can be used to block the q−1 lines parallel to this line (and thus construct the2
required blocking set), provided there are at least q− 1 of them. If ti > 1, we are done. If3
ti = 1, then either (q, t) = (3,4) or (q, t) = (4,5). The first case has been already excluded.4
In the second case, we have t1 = 1, t2 = t3 = 2. We change the direction of the 3 line5
segments in [x1] from [L1] to [L0]. Then each line in [L1] is blocked 6 times, while for [L2],6
[L3] we have t2 > 1, t3 > 1 and thus are done.7
〈ex:q=3t=4〉Example 4.10. The following construction produces a (48,4)-blocking set in the projective8
Hjelmslev planes PHG(Z39) and PHG(S3
3), with S33 = F3[X ]/(X2). The factor plane PG(2,3)9
contains an oval (quadrangle) which has 4 tangents and 6 external points (intersection points10
of the tangents). Each external point is on exactly two tangents. In each point class [x]11
external to the oval, place a double line segment in one of the two tangent directions. Choose12
the directions in such a way that no tangent is chosen more than twice. In each point class13
[x] on the oval, place a single line segment in the tangent direction. For those tangent14
directions [T ] which were chosen twice in the above process, arrange the 5 line segments in15
[T ] with direction [T ] in such a way that they block every line in [T ]. As is easily verified,16
the resulting point set forms a (48,4) blocking set. The complementary (69,8)-arc was17
originally discovered during a computer search [28]. The computational data suggested the18
preceding construction.19
Example 4.11. The general cascade construction.20
The following general cascade construction has been proposed in [22]. Let K0 be a21
(k0,n0)-arc in PG(2,q). Let suppK = {x1, . . . ,xk0} and let {X1, . . . ,Xk0} be a set of k0 lines22
in PG(2,q) such that xi ∈ Xi. Then for each pair of integers α,s ∈ {1, . . . ,q}, there exists an23
arc in PHG(R3R) with parameters (αsk0,max0≤i≤k0 νi), where ν0 = αn0 and νi = s+α|Xi∩24
suppK0|−α, for i = 1, . . . ,k0.25
Below a special instance of this construction is described. Take q2 = 25, s = 5, and K026
to be an (11,3)-arc in PG(2,5). There exist two such arcs and for both of them a1 +a2 = 11.27
Select the lines Xi to be the 1- and 2-lines of K0. It is easily checked that max0≤i≤k0 νi =28
max{5+α,3α}. For α = 2, we get a (110,7)-arc, while for α = 3, we get a (165,9)-arc.29
Example 4.12. Take K0 to be the trivial (q2 + q + 1,q + 1)-arc in PG(2,q) consisting of30
all the points of the plane. Index the point and line classes in PHG(R3R) in such a way31
that [xi] is incident with [Li] in the factor geometry, i = 1, . . . ,q2 + q + 1. Select a line32
segment in each neighbour class consisting of q points that are collinear with a line from33
[Li]. Denote this set of points by F . The arc χF has parameters (q(q2 + q + 1),2q). More34
importantly, it gives rise to a strongly regular graph in the following way (as described35
in [6]). Let C ≤ RRn, n = q(q2 + q + 1), be a linear code associated with F . Since every36
line of PHG(R3R) is incident with either q or 2q points of F , there are only two F -types37
of lines, (a0,a1,a2) = (q3,q2,q) and (q3,q2 − q,2q), which in turn yield three types of38
non-zero codewords in C , namely(a0(x),a1(x),a2(x)
)= (q3,q2,q), (q3,q2 − q,2q) and39
(0,q3,q2 + q) with corresponding frequencies q5 − q2, (q− 1)(q5 − q2) and q3 − 1. Now40
take G = (V,E) as the Cayley graph of (C ,+) with respect to the set C1 ⊂ C of codewords41
of type (q3,q2−q,2q), i.e. G has vertex set V = C and edge set E ={(x,y) | x−y ∈ C1
}.442
4Since C1 =−C1, this actually defines an undirected graph.
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As shown in [6], the graph G is strongly regular with parameters1
v = q6, k = q5−q2, λ = q4 +q3−3q2, µ = q4−q2.2
Moreover, C can be mapped (cf. [20]) onto a (possibly non-linear) two-weight code over3
Fq.4
In the next section, we give an algebraic construction for (k,2)-arcs.5
4.4 (k,2)-Arcs6
〈ssec:n=2〉For (k,2)-arcs we have the bound (9). In some cases this bound is achieved. There exists7
a (7,2)-arc in the plane over G2 = Z4, but there is no such arc in the plane over S2 =8
F2[x]/(x2). There exist (9,2)-arcs in the projective Hjelmslev planes over both chain rings9
with 9 elements. For larger chain rings, it is possible to get large (k,2)-arcs with more than10
one point in some of the neighbour classes.11
Remarkably, (q2 + q + 1,2)-arcs exist in the planes over the Galois rings G2r =12
GR(4r,4) for all r. Below, we explain the construction in a more general setting [14,23,24].13
Let q = pr > 1 be a prime power and Gq = GR(q2, p2) be the Galois ring of cardinality14
q2 and characteristic p2. For any k ∈ N, the ring Gqk is the unique Galois extension of Gq15
of degree k and conversely, Gqk contains a unique subring isomorphic to Gq. It is known16
that Gqk is free of rank k as a module over Gq. Hence, Gqk can be viewed as the underlying17
module of the (k−1)-dimensional projective Hjelmslev geometry over Gq. We denote this18
geometry by PHG(Gqk/Gq).19
The group G×q of units of Gq contains a unique cyclic subgroup Tq of order q−1, called20
the group of Teichmüller units. This applies to both Gq and its extension ring Gqk , and we21
have Tqk = 〈η〉, Tq = 〈η(qk−1)/(q−1)〉 for any element η ∈G×qk of order qk−1.22
?〈dfn:teich〉?Definition 4.13. The set {Gη j | 0 ≤ j < (qk −1)/(q−1)} is called the Teichmüller set of23
PHG(Gqk/Gq) and is denoted by Tq,k.24
Since {η j | 0 ≤ j < (qk − 1)/(q− 1)} is a set of coset representatives for Tq in Tqk ,25
the Teichmüller set Tq,k contains exactly one point from each neighbour class. In case of26
G2 = Z4, k odd, the linear code over Z4 associated with T2,k (via the columns of a generator27
matrix) is isomorphic to the shortened quaternary Kerdock code; cf. [13, 41].28
Recall that a set of points is called a cap if no three points of this set are collinear.29
〈thm:teichcap〉Theorem 4.14. Let Gq = GR(q2, p2) be a Galois ring of characteristic p2 and let k ≥ 3 be30
an integer.31
- If every prime divisor of k is larger than p, then the Teichmüller set Tq,k is a cap in32
PHG(Gqk/Gq).33
- If k is even, Tq,k is never a cap.34
In particular, the Teichmüller set T2r,3 forms a (22r +2r +1,2)-arc in the projective Hjelm-35
slev plane PHG(G23r/G2r)∼= PHG(G32r) over the Galois ring G2r .36
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For projective Hjelmslev planes over chain rings R containing a subring isomorphic to1
the residue field of R, the following result holds [23].2
〈thm:ovalimp1〉Theorem 4.15. Let R be a chain ring with |R| = 22r, R/N ∼= F2r , charR = 2. Then there3
exists no (22r +2r +1,2)-arc in the projective Hjelmslev plane PHG(R3R).4
At present, it is not known whether (22r +2r,2)-arcs do exist over chain rings of nilpo-5
tency index 2 and characteristic 2, except for the two smallest cases. The answer is positive6
for q = 2, but negative for q = 4; see [26].7
For odd characteristics, the following theorem has been recently proved [16].8
〈thm:ovalimp2〉Theorem 4.16. Let R = Fq[X ;σ]/(X2) be a chain ring of length 2 and prime characteristic.9
There exists a (q2,2)-arc in the projective Hjelmslev plane PHG(R3R).10
A (21,2)-arc in the plane over Z25 has been constructed recently [27]. Below, we give11
a (21,2)-arc in PHG(Z325) taken from the online tables [1]. The points are represented by12
the columns of a 3×21-matrix over Z25.13 0 1 5 1 1 15 1 1 10 1 1 1 1 1 1 0 1 1 1 1 10 5 1 7 15 1 0 3 1 11 18 24 2 13 22 1 1 20 4 12 141 0 6 17 24 4 1 3 18 7 15 7 8 22 11 15 11 23 1 24 3
14
4.5 Dual constructions15
〈ssec:dual〉Let Π = (P ,L , I) = PHG(R3
R) be a coordinate projective Hjelmslev plane over a finite16
chain ring R. Using duality properties of the inner product R3×R3 → R: (x,y) 7→ x · y =17
x1y1 + x2y2 + x3y3, one can show that the dual plane Π∗ = (L ,P , I∗) is isomorphic to the18
left coordinate plane PHG(RR3) or, what is the same, to the projective Hjelmslev plane19
PHG(S3S) over the opposite ring S = R◦. This duality can be exploited in some cases for20
new constructions of arcs with good parameters.21
〈ex:dualarcs〉Example 4.17. There exist maximal((q4 − q)/2,q2/2
)-arcs in the projective Hjelmslev22
planes over the Galois rings Gq, q = 2r. These arcs are obtained by taking K as the set of23
passants (0-lines) of a (q2 + q + 1,2)-arc in the corresponding dual plane. The new arcs24
have intersection numbers 0 and q2/2 with the lines of the dual plane and so are maximal.25
Since Gq = G◦q, the result follows.26
In the smallest case q = 2, the (7,2)-arc in PHG(Z34) is self-dual. In all other cases,27
Example 4.17 gives new arcs not covered by previous constructions, for example a (126,8)-28
arc in the plane over G4.29
?〈thm:dualimp〉?Theorem 4.18. Let R be a chain ring with |R|= 22r, R/N ∼= F2r , charR = 2. Then30
mq2/2(R3R)≤ q4/2−q/2−1. (11)31
Since (q2 +q +1,2)-arcs and((q4−q)/2,q2/2
)-arcs are dual to each other, this theo-32
rem is a corollary of Theorem 4.15.33
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4.6 Constructions using automorphisms1
?〈ssec:singer〉?By the Fundamental Theorem of Projective Hjelmslev Geometry [34], every collineation2
of a coordinate projective Hjelmslev plane Π = PHG(R3R) over a finite chain ring is in-3
duced by a semilinear automorphism of the underlying module R3R, and the collineation4
group of the plane PHG(R3R) is isomorphic to the projective semilinear group PΓL(3,R) =5
ΓL(3,R)/Z(R), where Z(R) denotes the center of the ring R.6
Automorphisms of Π can be used to considerably shorten searches for arcs with good7
parameters and make computer constructions of such arcs feasible which would otherwise8
be out of reach. As a simple example, we mention the fact that one can always assume9
the standard quadrangle (1,0,0)R, (0,1,0)R, (0,0,1)R, (1,1,1)R to be part of K, since10
PGL(3,R) acts regularly on ordered quadrangles in Π.11
The construction of discrete objects using incidence preserving group actions pioneered12
by Kerber et al. [3,25] can also be applied to the construction of arcs in projective Hjelmslev13
planes. To make the resulting computational tasks feasible for larger planes, one restricts14
attention to arcs which are invariant under certain automorphisms of Π, for example (lifted)15
Singer cycles of the factor plane PG(2,q). This method has been used successfully in [18,16
28] for the construction of new arcs with good parameters, accounting for many entries17
(lower bounds) in the tables of Section 4.7. The authors of [28] also maintain online tables18
of optimal arcs in projective Hjelmslev planes of small sizes [1].19
Suppose now that Π is a projective Hjelmslev plane over a Galois ring Gq, represented20
as PHG(Gq3/Gq) (cf. Section 4.4). A generator η of the Teichmüller subgroup Tq3 of G×q321
induces a collineation σ ∈ Aut(Π) of order q2 + q + 1, which acts as a Singer cycle on the22
factor plane PG(2,q). There is obviously a one-to-one correspondence between σ-invariant23
multisets in Π and multisets in a fixed point neighbour class of Π, for example [Gq1]. For24
a σ-invariant multiset K in Π, it is possible to compute the K-types of all lines in Π from25
certain combinatorial data of the corresponding multiset k in [Gq1] ∼= AG(2,q). As shown26
in [17], suitable choices of k yield σ-invariant arcs with good parameters. As an example27
of this construction, we mention a family of arcs in the planes over Gp, where p is an odd28
prime, which includes an optimal (39,5)-arc in the plane over Z9. A multiset k in AG(2, p)29
is called a triangle set if it is affinely equivalent to the set{(x,y) ∈ F2
p | x + y < p− 1}
.30
Here Fp = {0,1, . . . , p−1} is considered as a subset of Z.31
?〈thm:triangleset〉?Theorem 4.19 ( [17]). For every odd prime p, there exists a σ-invariant((p4− p)/2,(p2 +32
p)/2−1)-arc in the projective Hjelmslev plane over the Galois ring Gp. The arc is induced33
from an appropriately chosen triangle set in [Gp1]∼= AG(2, p).34
Finally we want to note that arcs in projective Hjelmslev planes with extremal param-35
eters may be of interest also from a group theoretic point-of-view (just like their classical36
counterparts). This is exemplified by the following result.37
?〈prop:zvenigorod〉?Proposition 4.20 ( [15]). The set H of hyperovals (maximal (7,2)-arcs) of PHG(2,Z4) has38
cardinality 256. The automorphism group G of PHG(2,Z4) acts transitively on H and the39
stabilizer Gh of a hyperoval h ∈ H has order 168. Furthermore, G has a normal subgroup40
H which acts regularly on H.41
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4.7 Tables for arcs in geometries over small chain rings1
〈ssec:tables〉 In the tables below, we summarize our knowledge about the values of mn(R3R) for the chain2
rings R with |R| = q2 ≤ 25, R/ radR ∼= Fq (cf. also [18]). We give information about all3
values of n with 2 ≤ n ≤ q2−1. The cases n = q2, . . . ,q2 +q are covered by Example 4.7.4
We want to remark the fact that we have lots of examples with mn(R3R) 6= mn(S3
S) for non-5
isomorphic chain rings R, S with |R| = |S|, R/ radR ∼= S/ radS (cf. Theorems 4.14, 4.156
and 4.16 and the results in Section 4.5). However, in all these examples charR 6= charS,7
and we do not have a single example of chain rings R and S of the same order, length and8
characteristic, in which the values of mn(R3R) and mn(S3
S) are different.9
n/R Z4 F2[X ]/(X2) Z9 F3[X ]/(X2)2 7 6 9 93 10 10 19 184 30 305 39 386 49 – 51 50 – 517 60 – 62 60 – 628 69 69
Table 1: Values of mn(R3R) for Hjelmslev planes of order q2 = 4 and q2 = 9
n/R G4 S4 T4
0 0 0 01 1 1 12 21 18 183 29 − 30 29 − 30 29 − 304 52 52 525 68 68 686 84 81 − 83 81 − 837 95 − 101 99 − 101 96 − 1018 126 120 − 125 120 − 1259 140 140 14010 152 − 160 152 − 160 152 − 16011 166 − 169 166 − 169 166 − 16912 186 − 189 186 − 189 186 − 18913 201 − 208 202 − 208 202 − 20814 224 − 228 216 − 228 219 − 22815 236 − 248 236 − 248 236 − 248
Table 2: Values of mn(R3R) for Hjelmslev planes of order q2 = 16
?〈tbl:q=4〉?
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n/R Z25 S5
0 0 01 1 12 21 253 40 − 43 42 − 434 66 − 70 64 − 705 85 − 102 90 − 1026 114 − 130 1307 142 − 156 152 − 1568 162 − 186 162 − 1869 186 − 208 190 − 208
10 210 − 238 225 − 23811 234 − 265 250 − 26512 260 − 295 280 − 295
n/R Z25 S5
13 310 − 311 297 − 31114 319 − 341 318 − 34115 355 − 367 355 − 36716 375 − 395 375 − 39517 400 − 425 405 − 42518 425 − 455 433 − 45519 465 − 466 455 − 46620 490 − 496 490 − 49621 515 − 525 515 − 52522 540 − 555 540 − 55523 565 − 585 565 − 58524 595 − 615 595 − 615
Table 3: Values of mn(R3R) for Hjelmslev planes of order q2 = 25
?〈tbl:q=5〉?
5 Blocking sets in projective Hjelmslev planes1
?〈sec:blocking〉?5.1 General results2
?〈ssec:general〉??〈dfn:blockingsets〉?Definition 5.1. A multiset K in (P ,L , I) is called a (k,n)-blocking multiset if3
(i) K(P ) = k;4
(ii) K(L)≥ n for every line L ∈ L .5
Similarly to Definition 4.1, we assume in addition that there exists at least one line L06
with K(L0) = n. A (k,n)-blocking multiset K is called minimal if it does not contain a7
(k−1,n)-blocking multiset, i.e. decreasing the multiplicity of any point p ∈ suppK by one8
yields a multiset K′ with K′(L) = n− 1 for some line L ∈ L . Blocking sets (i.e. projective9
blocking multisets) and projective arcs are complementary concepts in the sense that the10
complement of a projective (k,n)-arc in P is a (q4 + q3 + q2 − k,q2 + q− n)-blocking set11
and vice versa.12
First, let us consider blocking sets in planes over general chain rings R with |R| = qm,13
R/N ∼= Fq. For (k,n)-blocking sets in such planes, we have the following theorem [36].14
〈thm:bsets1〉Theorem 5.2. Let R be a chain ring with |R| = qm, R/N ∼= Fq, and let K be a (k,n)-15
blocking multiset with 1 ≤ n ≤ q, in Π = PHG(R3R). Then k ≥ nqm−1(q + 1). If K is a16
(k,n)-blocking multiset with k = nqm−1(q+1), n < q/p, where p = charFq, then there exist17
lines, L1,L2, . . . ,Ln say, such that18
K(1)([x])= qm−1|{ j | j ∈ {1, . . . ,n},([x], [L j]) ∈ J(1)}|.19
The second part of the theorem says that the induced multiset K(1)/qm−1 is a sum of20
lines. It is impossible to generalize this to the stronger condition: “K(i)/qm−i is a sum of21
lines for some i > 1”.22
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For the most interesting case of (k,1)-blocking sets, we have k ≥ qm−1(q + 1) and1
in case of equality the support of such a blocking set is necessarily a line. By taking a2
line L and from each class [x]m−1 incident with [L]m−1 in (P (m−1),L(m−1),J(m−1)) exactly3
n− 1 further line segments in the direction of L, one obtains for each n ∈ {1,2, . . . ,q} an4 (n,nqm−1(q + 1)
)-blocking set, showing that the extremal cases k = nqm−1(q + 1) of The-5
orem 5.2 can be realized by projective multisets.6
Under certain conditions, some subplanes of PHG(R3R) form a blocking set.7
?〈thm:subplane〉?Theorem 5.3. Let R be a chain ring with |R|= qm, R/N ∼= Fq, where qm is a perfect square.8
Let there exist a subring S of R that is a chain ring with |S| = qm/2 and such that R is free9
over S. Then the multiset K defined by10
K(x) ={
1 if x is a point from PHG(S3S),
0 otherwise,11
is a blocking set in PHG(R3R).12
In the special case when R is a chain ring with |R| = q2, R/N ∼= Fq, that contains a13
subring S isomorphic to the residue field Fq, PHG(R3R) contains a subplane Π′ isomorphic14
to PG(2,q) and the projective multiset K defined by suppK = Π′ is an irreducible (q2 +q+15
1,1)-blocking set. These blocking sets are introduced in [5] in a slightly different context.16
They are defined as the orbit of a fixed point with coordinates from the field Fq under a17
Singer cycle of PG(2,q). As shown in [6], linear codes associated with these multisets can18
be mapped (cf. [20]) to two-weight linear codes over Fq. These in turn give rise to a family19
of strongly regular graphs with parameters20
v = q6, k = q4−q, λ = q3 +q2−3q, µ = q2−q.21
Let us now consider planes over chain rings with |R| = q2, R/N ∼= Fq. It is of interest22
to find the smallest size of a minimal blocking set which is not a line. Unlike the situation23
in the classical projective planes where there is a gap between the size of a line and the size24
of the smallest non-trivial blocking sets (see e.g. [4]), there exist minimal blocking sets of25
size q2 +q+1 in all planes PHG(R3R).26
?〈thm:nonexbaer2〉?Theorem 5.4. Let K be a minimal (q2 + q + 1,1)-blocking set in PHG(R3R), |R| = q2,27
R/ radR ∼= Fq. Then K is of one of the following types:28
(1) a projective plane of order q;29
(2) for lines L0 and L1 with L0 _ L1, and a point z ∈ L0 \L130
K(x) ={
1 if x ∈ (L0 \ [z])∪{z} or x ∈ L1∩ [z]0 otherwise.
(12)31
If R = GR(q2, p2), then there is no (q2 +q+1,1)-blocking set of type (1).32
Let us note that the blocking set described in (12) is in some sense trivial since K(1) =33
q · χ[L] + χ[z] consists of a q-fold line and a further point on this line. We would like to34
construct non-trivial blocking sets also for the planes over the Galois rings Gq. This can35
be done by generalizing the familiar technique of Rédei type blocking sets to projective36
Hjelmslev planes.37
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5.2 Rédei type blocking sets1
?〈ssec:redei〉?As before let Π = PHG(R3R), where R is a chain ring of nilpotency index 2. Fix a generator θ2
of radR and a set Γ⊂ R of representatives for the residue classes in R/ radR∼= Fq. Suppose3
that Γ = {γ0,γ1, . . . ,γq−1} with γ0 = 0, γ1 = 1, and hence radR = {γiθ | 0 ≤ i ≤ q− 1} =4
{θγ j | 0≤ j ≤ q−1}. Thus each c ∈ radR has unique representations c = γiθ = θγ j, where5
in the non-commutative cases i, j may be different.6
As already noted, the affine plane AHG(R2R) is obtained by deleting a neighbour class7
of lines (the “class at infinity”) together with all points incident with a line in this class.8
With no loss of generality we can take the class [z = 0] as the class at infinity. This class9
consists of all lines with equations of the form aX + bY + Z = 0, where a,b ∈ radR. All10
points incident with lines in this class have homogeneous coordinates (x,y,z) with z∈ radR.11
The points outside this class have coordinates (x,y,1), x,y ∈ R. Now the points of the affine12
plane AHG(R2R) are identified with the pairs (x,y), where x,y ∈ R. The lines of AHG(R2
R)13
have equations Y = aX + b or X = cY + d, a,b,d ∈ R, c ∈ radR. We say that a line of the14
first type has slope a. A line with equation X = cY +d is said to have slope ∞ j, if c = θγ j,15
j = 0,1, . . . ,q−1.16
The infinite points on a fixed line L from the neighbour class of infinite lines can be17
identified with the slopes. So, (a) (resp. (∞ j)) will denote the infinite point from L of the18
lines with slope a (resp. ∞ j). The q2 lines with a fixed slope form a parallel class of lines19
in AHG(R2R), and the line set of AHG(R2
R) is partitioned into q2 +q such parallel classes.20
Definition 5.5. Let U be a set of q2 points in AHG(R2R). We say that the infinite point (a)21
is determined by U if there exist different points u,v ∈U such that u,v and (a) are collinear22
in PHG(R3R).23
Note that in view of the assumption |U |= q2, the point (a) is determined by U iff there24
exists a line in AHG(R2R) with slope a which is disjoint from U .25
?〈thm:main〉?Theorem 5.6. Let U be a set of q2 points in AHG(R2R). Denote by D the set of infinite points26
determined by U and by D(1) the set of neighbour classes on the infinite line containing27
points from D. If |D|< q2 +q, then there exists a minimal blocking set in PHG(R3R) of size28
q2 + q + 1 + |D|− |D(1)| that contains U. In particular, if D contains representatives from29
all neighbour classes on the infinite line, then B = U ∪D is a minimal blocking set of size30
q2 + |D| in PHG(R3R).31
The above construction gives blocking sets of size at most 2q2 +q−1. We are interested32
in sets U that are of the form33
U = {(x, f (x)) | x ∈ R}34
for some suitably chosen function f : R→ R. Let x and y be two different elements from R.35
We have the following possibilities:36
1) if x− y 6∈ radR, then (x, f (x)) and (y, f (y)) determine the point (a), where37
a = ( f (x)− f (y))(x− y)−1.38
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162 T. Honold and I. Landjev
2) if x−y∈ radR\{0}, and f (x)− f (y) 6∈ radR, the points (x, f (x)) and (y, f (y)) determine1
the point (∞ j) if2
(x− y)( f (x)− f (y))−1 = θγ j, γ j ∈ Γ.3
3) if x−y ∈ radR\{0}, and f (x)− f (y) ∈ radR, say x−y = θa, f (x)− f (y) = θb, a,b ∈ Γ4
and5
a) if b 6= 0, then (x, f (x)) and (y, f (y)) determine all points (c) with c ∈ ab−1 + radR;6
b) if b = 0, then (x, f (x)) and (y, f (y)) determine the infinite points (∞0), . . . ,(∞q−1).7
Example 5.7. Let R be a chain ring with |R| = q2, R/ radR ∼= Fq that contains a proper8
subring isomorphic to its residue field Fq (i.e. one of the rings Sq or T(i)q ).9
Define10
f : R → R : a+θb 7→ b+θa. (13)11
It can be checked that the set of points U = {(x, f (x)) | x ∈ R} determines q + 1 infinite12
points.13
We can compute the parameters of the Rédei-type blocking sets given by (13) also for14
the plane over the Galois ring Gq = GR(q2, p2). In this case, U determines exactly q2−q+215
directions, and the size of the corresponding Rédei-type blocking set is 2q2−q+2.16
Below we will give two further of examples Rédei-type blocking sets in the plane over17
Gq. For these examples, we need to collect a few additional facts about Gq.18
In the case of Gq (and Galois rings in general), there are canonical choices for θ and19
Γ, which we will adopt for the rest of this paper. Since radGq = pGq, we can take θ = p20
as a generator of radGq. Furthermore, since the augmented Teichmüller subgroup Γq :=21
Tq∪{0} (for the definition of Tq, see Section 4.4) forms a system of coset representatives22
modulo radGq, we can take Γ = Γq.23
Every a ∈Gq can be written in exactly one way as a = a0 +a1 p with a0,a1 ∈ Γq.524
〈fact:witt〉Fact 5.8. The ring Gq is isomorphic to the ring W2(Fq) of so-called Witt vectors of length2 over Fq, which is defined as the set of all pairs (a,b)∈ Fq×Fq with the following additionand multiplication:
(a0,a1)+(b0,b1) = (a0 +b0,a1 +b1−p−1
∑j=1
1p
(pj
)a j
0bp− j0 ),
(a0,a1) · (b0,b1) = (a0b0,ap0b1 +bp
0a1).
The map φ : Gq →W2(Fq) : a0 + a1 p 7→ (a0,a1p), where a = a + radGq, provides a ring25
isomorphism.26
For the definition of Witt vectors see [45], and for a proof of 5.8 see [43]. Working with27
Witt vectors instead of the original representation of Gq = Zp2 [X ]/(h) has the advantage28
that all computations are now done in Fq.29
5This is true regardless of the particular choice of Γ. However, for the following Fact 5.8 the choice Γ =Tq∪{0} is essential.
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Example 5.9. Let q = pr, where p is odd. We are going to define f as a function on W2(Fq).1
For x = (a0,a1), set2
f (x) ={
(a0,a1) if a0 is a square in Fq,(−a0,−a1) if a0 is a non-square in Fq.
(14)3
?〈thm:square〉?Theorem 5.10. Let R = GR(q2, p2), q = pm, p odd. The set U = {(x, f (x)) | x ∈ R}, where4
the function is defined in (14), determines5
q2
2+
32
q6
directions in AHG(R2R). Furthermore, there exists a Rédei type blocking set in PHG(R3
R) of7
size8
32
q2 +2q− 12.9
In our last example, we will construct a Rédei type blocking set over the Galois ring10
S = Gqm , where m ≥ 1 is arbitrary, using the fact that S is a Galois extension of R = Gq.11
Recall that the trace function TrS/R : S → R is defined as12
TrS/R(x) := ∑σ∈Aut(S/R)
σ(x) =m−1
∑i=0
(xqi
0 + xqi
1 p) for x ∈ S, (15)13
where x = x0 + x1 p with x0,x1 ∈ Γqm .14
Example 5.11. As above let R = Gq and S = Gqm . We define a Rédei type blocking set in15
PHG(S3S) by setting f (x) = TrS/R(x).16
?〈thm:trace〉?Theorem 5.12. Let R = GR(q2, p2) and let S be an extension of R of degree m. The set17
U = {(x, f (x)) | x ∈ S} defined by the function f (x) = TrS:R(x) determines18
qm−1q−1
qm19
directions in AHG(S2S). There exists a Rédei type blocking set in PHG(S3
S) of size20
q2m +qm +1+qm−1q−1
qm−qm−1.21
Acknowledgement22
The authors wish to thank Michael Kiermaier for help with the tables in Section 4.7 and23
with Examples 4.10 and 4.17.24
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