fuchsian codes with arbitrary rates - ghent universitycage.ugent.be/cost/talks/blanco-chacon.pdf ·...
TRANSCRIPT
Fuchsian codes with arbitrary rates
Iván Blanco Chacón, Aalto University
20-09-2013
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Summary
Basic concepts in wireless channels
Arithmetic Fuchsian groups: the rational case
Arithmetic Fuchsian groups: the general case
Fundamental domains and point reduction
Fuchsian codes with rate 3
Fuchsian codes with arbitrary rate
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Basic concepts on Wireless channels
An Additive White Gaussian Noise (AWGN) is a sequence of
independent (in time) iidd complex Gaussian random variables. A
wireless channel a�ected only by AWGN has a channel equation
Y = X + N where X is a transmitted vector, N ∼ CN(0,Σ) and Y
the received vector.
We suppose that the AWGN originates at the receiving end.
A fading wireless channel is a channel subject to interference and
eventually to AWGN. It has a channel equation Y = HX + N,
where H is a fading matrix and Y ,X ,N as before.
We suppose that H is perfectly known to the receiver.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Basic concepts on Wireless channels
A QAM constellation is a set of the form
{(2n + 1, 2m + 1) : 0 ≤ |n|, |m| ≤ M}. It represents M2 di�erent
transmission states. It requires a labeling algorithm.
More generally, one has NUPAM and NUQAM alphabets,
non-uniformly distributed.
We are interested to send several NUPAM symbols simultaneously
by a single fading channel (SIMO), and to obtain a low complexity
decoding method.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Basic concepts on Wireless channels
How to send QAM/NUQAM symbols by a fading channel? Without
coding�>Need to solve: Let C be a constellation, suppose you
receive y , minimize
minx∈C ||y − Hx ||
Too complex by brute force if |C | >> 1!. Alternative: extra
structure on the codebook (lattices, matrix orders) . But how to
compare codes? Complexity and SNR/BEP.
SNR(Signal to noise ratio) = Energy/||Σ||
Diagrams BEP/SNR
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Basic concepts on Wireless channels
æ
æ
æ
æ
æ
à
à
à
à
à
à
ì
ì
ò
ò
ò
ò
ò
ò
ô
ô
ô
ô
ô
ô
ô
ô
12 14 16 18 20
10-4
0.001
0.01
ô 4-G@15,1Dò 4-G@10,1Dì 4-G@6,1Dà 4-G@g=1Dæ 4-QAM
Linear codes coming from cyclic division algebras have complexity
O(|C |r ), with r ∈ Q, typically r = 1/2 (Alamouti) or r = 0,625
(Golden). Our approach is non-linear and yields complexity
O(log |C |).
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Arithmetic Fuchsian groups: the rational case
a, b ∈ Q∗; B =(a,bQ
)= Q + QI + QJ + QK
I 2 = a; J2 = b, IJ = −JI = K
Reduced norm: N(x + yI + zJ + tK ) = x2 − ay2 − bz2 + abt2
Reduced Trace: Tr(x + yI + zJ + tK ) = 2x
ψ :
(a, b
Q
)↪→ M
(2,Q(
√a))
x + yI + zJ + tK 7→
(x + y
√a z + t
√a
b(z − t√a) x − y
√a
)
If B ⊗Qp is a division algebra, we say that B rami�es at p,
p prime or ∞; Q∞ = R.
D(B)=product of the rami�cation primes of B .
If B rami�es at p =∞, it is said to be de�nite; otherwise inde�nite.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Arithmetic Fuchsian groups: the rational case
An order O of B is a Z-lattice such that O ⊗Z Q ∼= B and such
that O is also a ring.
B =
(a, b
Q
)inde�nite; OB maximal order (up to conjugation)
O1
B = multiplicative group of elements reduced norm 1
Γ1B = ψ(O1
B)
If B is inde�nite of discriminant D, denote Γ(D, 1) := Γ1B
An arithmetic Fuchsian group of the �rst kind is a discrete group
Γ ⊆ GL (2,R) commensurable with Γ1B for some B .
Examples: Γ0(N), Takeuchi groups.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Arithmetic Fuchsian groups: the general case
F = Q(θ) totally real number �eld of degree n; RF its ring of
integers; a, b ∈ F ∗;
A quaternion F -algebra is B =(a,bQ
)= Q + QI + QJ + QK such
that I 2 = a; J2 = b, IJ = −JI = K .
An order O of B is a ring which is a rank 4 RF -lattice.
condition S: B rami�es EXACTLY at one absolute value in F
extending the usual one in Q.
Condition S allows us to assume that a > 0 so that we have again a
representation
ψ :
(a, b
Q
)↪→ M
(2,F (
√a))
x + yI + zJ + tK 7→
(x + y
√a z + t
√a
b(z − t√a) x − y
√a
)Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fundamental domains and point reduction
If Γ is an arithmetic Fuchsian group, then Γ acts on H. Afundamental domain is F such that
for any z ∈ H, there exists w ∈ F and g ∈ Γ such that
g · z = w ,
for any z ,w ∈ F and g ∈ Γ such that g · z = w , z ,w ∈ Fr(F).
Arithmetic Fuchsian groups have nice fundamental domains. They
tessellate H and we will use them for our coding purposes.
Condition S and rami�cation in prime ideals implies that F is
compact.
Let us see some examples:
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fundamental domains and point reduction
Figura : Fundamental domain for SL (2,Z)
S =
(0 1
−1 0
), T =
(1 1
0 1
), SL (2,Z) = 〈S ,T 〉
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fundamental domains and point reduction
Arithmetic Fuchsian group of signature (1, e):
Γ = 〈α, β|[α, β]e = ±1〉, α =
(λ 0
0 λ−1
), β =
(2 1
1 2
)(λ
algebraic number).
-2 -1 0 1 2
-2
-1
0
1
2
Figura : Fundamental domain for signature (1, 2)Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fundamental domains and point reduction
-2 -1 0 1 2
1
2
3
4
Figura : Fundamental domain for signature Γ(6, 1) and tessellation
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fundamental domains and point reduction
Problem 1: How to �nd presentations for an arithmetic Fuchsian
group? This is equivalent to the problem of how to produce
fundamental domains.(Bayer, Alsina, Voight)
Problem 2: Given a presentation of an arithmetic Fuchsian group,
decompose matrices as products of the generators. Equivalent to
the problem of given a fundamental domain F and a point outside
it, to �nd a transformation which brings the point inside F . (Bayer,B., Remón).
The following result implies the low decoding complexity of our
codes:
Theorem (Bayer-B. 2012, Bayer-Remón 2013)
Given a fundamental domain for an arithmetic Fuchsian group,
there exists an explicit point reduction algorithm doing at most as
many matrix products as the minimal length of the input matrix.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes of rate 3
Alsina and Bayer have shown:
Γ(6, 1) ={γ =
1
2
(α β
−β′ α′
)|α, β ∈ Z[
√3], det(γ) = 1, α ≡ β mod 2
}.
Alternatively Γ(6, 1) = 〈γ1, γ2, S〉, γ1 = 1
2
(1 +√3 −3 +
√3
3 +√3 1−
√3
)
and γ2 = 1
2
(1 +√3 3−
√3
−3−√3 1−
√3
).
Consider in Γ(6, 1) the subgroup given by the group of units of
norm 1 in the natural order Z[1, I , J,K ]. Since α, β are restricted to
the determinant condition, they carry 3 degrees of freedom over Z.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes of rate 3
Suppose one Tx -transmission antenna- and one or more Rx
-receiving antennas-
Suppose a �nite codebook C and a �nite collection of 4-tuples of
integers {(xi , yi , zi , ti )}|C |i=1such that x2i − ay2i − bzi + abti = 1.
Want to summarize each 4-tuple into a suitable signal (coding),
send this signal by the antenna, and decode it. Each 4-tuple will
give a matrix belonging to an arithmetic Fuchsian group of the �rst
kind attached to B =(a,bQ
)(x , y , z , t) γ =
(x + y
√a z + t
√b
−z + t√b x − y
√a
)
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes of rate 3
Fix a fundamental domain F for Γ. We will send γ(τ), where
τ ∈ F is an interior point.
Interior points are Γ-inequivalent, so we will have as many symbols
as codewords.
Design Problem number 1: How to choose τ?
Want to transmit over a fading channel. But assume for simplicity
just AWGN.
KEY IDEA: Transmit γ(τ). It belongs to γ(F). It is an interior
point. For any interior point w ∈ γ(F), the reduction algorithm
returns a representative z ∈ F and the unique transformation that
brings z into w . Suppose that the fundamental domain, the
euclidean center τ of it and the AWGN is such that the BEP is very
very small. Then, the reduction algorithm returns ±γ for γ(τ) + N.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes of rate 3
Some simulations:
æ
æ
æ
æ
æ
à
à
à
à
à
à
ì
ì
ò
ò
ò
ò
ò
ò
ô
ô
ô
ô
ô
ô
ô
ô
12 14 16 18 20
10-4
0.001
0.01
ô 4-G@15,1Dò 4-G@10,1Dì 4-G@6,1Dà 4-G@g=1Dæ 4-QAM
Figura : 4NUF constellations
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes of rate 3
Some simulations:
æ
æ
æ
æ
æ
æ
à
à
à
à
à
à
ì
ì
ì
ì
ì
ì
ì
ìì
ì
ò
ò
ò
ò
ò
ò
ô
ô
ô
ô
ô
ô
ôô
ôô
ôôôôô
15 20 25 30
10-5
10-4
0.001
0.01
0.1
ô 8-G@15,1Dò 8-G@10,1Dì 8-G@6,1Dà 8-G@g=1Dæ 8-QAM
Figura : 8NUF constellations
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes of rate 3
Some simulations:
æ
æ
æ
æ
æ
à
à
à
à
à
à
àà
àà
àààà
ì
ì
ì
ì
ì
ìì
ìì
ììììì
ò
ò
ò
ò
ò
ò
òò
òò
òòò
ô
ô
ô
ô
ôô
ôô
ôô
ôô
ôôôôô
15 20 25 30
10-5
10-4
0.001
0.01
0.1
1
ô 16-G@15,1Dò 16-G@10,1Dì 16-G@6,1Dà 16-G@g=1Dæ 16-QAM
Figura : 16NUF constellations
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes with arbitrary rate
F = Q(θ)/Q totally real of degree n; B =(a,bF
)meeting condition
S ; �x an RF -order O (for simplicity, the natural one). We can
regard Γ := ψ(O)1 as matrices(x +√ay z +
√bt
b(z −√at x −
√ay)
); x , y , z , t ∈ RF .
Since x =∑n−1
k=0mkθ
k , with mk ∈ Z and analogously y , z , t, the
matrices in M(2,RF [
√a])carry 4n items of information.
New proposal: Fix F for Γ. Send a 4-tuple (x , y , z , t) satisfying
x2 − ay2 − bz2 + abt2 = 1 as a matrix acting on τ ∈ F , γ(τ).
Send γ(τ) by an AWGN channel and use the reduction point
algorithm to decode.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Fuchsian codes with arbitrary rate
Proposition
The code rate of the proposed code (using N channels) is 3n/N.
Example 1: Take B =(
3,−1Q(√7)
)and the natura order
Z[√7][1, I , J,K ]. The corresponding Fuchsian code has rate 6. The
matrix
( √7 +√3
√3√
3√7−√3
)is identi�ed with the 8-tuple
(0, 1, 1, 0, 0, 0, 1, 0).
Example 2: Take F the maximal totally real sub�eld of the p-th
cyclotomic �eld. And a quaternion F -algebra meeting condition S
(p = 13 works, we think that in�nitely many). Then, the Fuchsian
code has rate 3(p−1)2
.
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates
Bibliography
M. Alsina, P. Bayer: Quaternion orders, quadratic forms and
Shimura curves. CRM Monograph Series, 22. American
Mathematical Society, Providence, RI 2004.
M. Alsina, I. Blanco-Chacón, D. Remón, C. Hollanti: Fuchsian
codes for AWGN channels (extended journal version).
Submited.
I. Blanco-Chacón, D. Remón, C. Hollanti: Fuchsian codes for
AWGN channels. Proceedings of the International Workshop in
Cryptography and Coding WCC2013, 496-507.
F. E. Oggier, J.C. Bel�ore, E. Viterbo: Cyclic division algebras:
A tool for space-time coding. Foundations and Trends in
Communications and Information Theory, vol. 4, no. 1 (2007).
Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates