coding for the permutation channel - universiteit gent

Coding for the Permutation Channel

Mladen Kovačevid

(joint work with Dejan Vukobratovid)

Department of Power, Electronics, and Communication Engineering, University of Novi Sad, Serbia

Summary

• Permutation channels

• Error correction in permutation channels

– Multiset codes

– Perfect multiset codes

2

Permutation channel

3

Permutation channel

• Channel alphabet: 𝑆, a finite set of symbols

3

Permutation channel


• Channel input: Sequence of symbols 𝒄 = 𝑐1, 𝑐2, … , 𝑐𝑛 , 𝑐𝑖 ∈ 𝑆

3

Permutation channel



• Channel output: Random permutation of 𝒄, namely

𝒄 = 𝑐𝜋 1 , 𝑐𝜋 2 , … , 𝑐𝜋 𝑛

3

Permutation channel




𝒄 = 𝑐𝜋 1 , 𝑐𝜋 2 , … , 𝑐𝜋 𝑛

• Equivalently: 𝒄 = 𝒄 ⋅ Π

• where Π is a random 𝑛 × 𝑛 permutation matrix

3

Permutation channel




𝒄 = 𝑐𝜋 1 , 𝑐𝜋 2 , … , 𝑐𝜋 𝑛

• Equivalently: 𝒄 = 𝒄 ⋅ Π

• where Π is a random 𝑛 × 𝑛 permutation matrix

3

– This is a special case of the “random matrix” channel arising in random linear network coding

Permutation channel

• Example: End-to-end transmission in some types of packet networks

4

Permutation channel


– Multipath routing ⇒ reordering of packets

4

Permutation channel



– Symbols = packets

4

Permutation channel




• Apart from being permuted, some packets can be lost, some can be received erroneously, etc.

4

Permutation channel




• Apart from being permuted, some packets can be lost, some can be received erroneously, etc.

• To account for these impairments, we model the channel as a permutation channel with errors, deletions, and insertions.

4

Coding for the permutation channel

5


• In the permutation channel, any information that is contained in the order of symbols will be lost

5



– Example: Given the received sequence, say (𝑐, 𝑎, 𝑏), any of the following sequences is equally likely to have been sent: 𝑎, 𝑏, 𝑐 , 𝑎, 𝑐, 𝑏 , 𝑏, 𝑎, 𝑐 , 𝑏, 𝑐, 𝑎 , 𝑐, 𝑎, 𝑏 , (𝑐, 𝑏, 𝑎)

5




• The only carrier of information should be the (multi)set of symbols selected at the source

5




• The only carrier of information should be the (multi)set of symbols selected at the source

– Error-correcting codes for this channel should be defined in the space of all multisets over the channel alphabet 𝑆

5

Multiset codes

6

Multiset codes

• Recall: A multiset is a set with repetitions of elements allowed

6

Multiset codes


– Operations on multisets:

• *𝑎, 𝑏, 𝑏+ = 3

• *𝑎, 𝑏, 𝑏+ ∪ *𝑎, 𝑎, 𝑏, 𝑐+ = *𝑎, 𝑎, 𝑏, 𝑏, 𝑐+

• *𝑎, 𝑏, 𝑏+ ∩ *𝑎, 𝑎, 𝑏, 𝑐+ = *𝑎, 𝑏+

6

Multiset codes


– Operations on multisets:

• *𝑎, 𝑏, 𝑏+ = 3

• *𝑎, 𝑏, 𝑏+ ∪ *𝑎, 𝑎, 𝑏, 𝑐+ = *𝑎, 𝑎, 𝑏, 𝑏, 𝑐+

• *𝑎, 𝑏, 𝑏+ ∩ *𝑎, 𝑎, 𝑏, 𝑐+ = *𝑎, 𝑏+

• Notation:

– 𝑀 𝑆 = the set of all multisets over 𝑆

– 𝑀 𝑆, 𝑛 = the set of all multisets over 𝑆 of cardinality 𝑛

6

Multiset codes

• Definition: A multiset code 𝐶 is a non-empty subset of 𝑀(𝑆). We say that it is a constant-cardinality code if 𝐶 ⊆ 𝑀(𝑆, 𝑛).

7

Multiset codes


• Natural metrics in 𝑀(𝑆) are:

𝐷 𝑋, 𝑌 = 𝑋 ∪ 𝑌 − 𝑋 ∩ 𝑌 = 𝑋 ∖ 𝑌 + 𝑌 ∖ 𝑋

7

Multiset codes



𝐷 𝑋, 𝑌 = 𝑋 ∪ 𝑌 − 𝑋 ∩ 𝑌 = 𝑋 ∖ 𝑌 + 𝑌 ∖ 𝑋

𝐷′ 𝑋, 𝑌 = max 𝑋 ∖ 𝑌 , 𝑌 ∖ 𝑋

7

Multiset codes



𝐷 𝑋, 𝑌 = 𝑋 ∪ 𝑌 − 𝑋 ∩ 𝑌 = 𝑋 ∖ 𝑌 + 𝑌 ∖ 𝑋

𝐷′ 𝑋, 𝑌 = max 𝑋 ∖ 𝑌 , 𝑌 ∖ 𝑋

• Claim: A multiset code 𝐶 with minimum distance 𝐷min is capable of correcting any pattern of 𝑠 insertions, 𝜌 deletions, and 𝑡 substitutions (errors), as long as 2 𝑠 + 𝜌 + 2𝑡 < 𝐷min.

7

Multiset codes as integer codes

9


• Claim: Multiset codes are equivalent to integer codes under ℓ1 distance

9



• Every multiset over 𝑆 can be described by its multiplicity function

9




– Example: Let 𝑆 = *𝑎, 𝑏+, and observe the multiset 𝑋 = *𝑎, 𝑎, 𝑎, 𝑏, 𝑏+. We can simply describe it as (3, 2), i.e., 3 𝑎’s and 2 𝑏’s.

9




– Example: Let 𝑆 = *𝑎, 𝑏+, and observe the multiset 𝑋 = *𝑎, 𝑎, 𝑎, 𝑏, 𝑏+. We can simply describe it as (3, 2), i.e., 3 𝑎’s and 2 𝑏’s.

– Example: Let 𝑆 = *𝑎, 𝑏, 𝑐+, and observe the multiset 𝑋 = *𝑏, 𝑏, 𝑐, 𝑐+. We can simply describe it as (0, 2, 2), i.e., 0 𝑎’s, 2 𝑏’s, and 2 𝑐’s.

9


• If 𝑆 = *0, 1,… , 𝑞 − 1+, then every multiset 𝑋 over 𝑆 can be

described as a 𝑞-tuple 𝒙 = 𝑥0, … , 𝑥𝑞−1 , 𝑥i ∈ ℕ

10




– 𝑥𝑖’s are non-negative integers representing numbers of occurrences of the symbols from 𝑆 in 𝑋

10





– The set of all multisets over 𝑆 is equivalent to ℕ𝑞

10





– The set of all multisets over 𝑆 is equivalent to ℕ𝑞

• We have:

– 𝑋 ∪ 𝑌 ↔ max 𝒙, 𝒚 (componentwise)

– 𝑋 ∩ 𝑌 ↔ min 𝒙, 𝒚 (componentwise)

– 𝑋 ↔ 𝑥𝑖𝑖

10


• Also, the distance between multisets 𝑋 and 𝑌 reduces to the ℓ1 (Manhattan) distance between their multiplicity functions 𝒙, 𝒚 ∈ ℕ𝑞

11



𝐷 𝑋, 𝑌 = 𝒙 − 𝒚 1 = 𝑥𝑖 − 𝑦𝑖

𝑞−1

𝑖=0

11



𝐷 𝑋, 𝑌 = 𝒙 − 𝒚 1 = 𝑥𝑖 − 𝑦𝑖

𝑞−1

𝑖=0

• Therefore, multiset codes are equivalent to codes in ℕ𝑞 under the ℓ1 metric

11



𝐷 𝑋, 𝑌 = 𝒙 − 𝒚 1 = 𝑥𝑖 − 𝑦𝑖

𝑞−1

𝑖=0

• Therefore, multiset codes are equivalent to codes in ℕ𝑞 under the ℓ1 metric

11

• Constant-cardinality codes are equivalent to codes in the “discrete simplex”

Δ𝑛𝑞−1= 𝑥0, … , 𝑥𝑞−1 ∶ 𝑥𝑖 ∈ ℕ, 𝑥𝑖

𝑖= 𝑛

Special case – Subset codes

12


• If we restrict to sets, we get subset codes, i.e., codes in the power set of 𝑆

𝑃 𝑆 = 𝑋 ∶ 𝑋 ⊆ 𝑆

12



𝑃 𝑆 = 𝑋 ∶ 𝑋 ⊆ 𝑆

• These codes are equivalent to binary codes in the Hamming space, whereas constant-cardinality codes are equivalent to constant-weight binary codes (codes in the Johnson space)

12



𝑃 𝑆 = 𝑋 ∶ 𝑋 ⊆ 𝑆


• Remark: The length and minimum distance of subset codes are bounded by 𝑆

12



𝑃 𝑆 = 𝑋 ∶ 𝑋 ⊆ 𝑆



– the same “problem” appears with subspace codes

12



𝑃 𝑆 = 𝑋 ∶ 𝑋 ⊆ 𝑆



– the same “problem” appears with subspace codes

– Multiset codes do not have this restriction; 𝑀(𝑆) is infinite

12

Perfect multiset codes

13


• Let 𝑈, 𝑑 be a metric space with an integer-valued metric 𝑑. A code 𝐶 ⊆ 𝑈 is said to be 𝑒-perfect, 𝑒 ∈ ℕ, if

13



𝐵 𝒙, 𝑒 ∩ 𝐵 𝒚, 𝑒 = ∅, 𝒙, 𝒚 ∈ 𝐶, 𝒙 ≠ 𝒚

• and

13



𝐵 𝒙, 𝑒 ∩ 𝐵 𝒚, 𝑒 = ∅, 𝒙, 𝒚 ∈ 𝐶, 𝒙 ≠ 𝒚

• and

𝐵 𝒙, 𝑒

𝒙∈𝐶

= 𝑈

13



𝐵 𝒙, 𝑒 ∩ 𝐵 𝒚, 𝑒 = ∅, 𝒙, 𝒚 ∈ 𝐶, 𝒙 ≠ 𝒚

• and

𝐵 𝒙, 𝑒

𝒙∈𝐶

= 𝑈

• where 𝐵 𝒙, 𝑒 = 𝒘 ∶ 𝑑 𝒙,𝒘 ≤ 𝑒 .

13



𝐵 𝒙, 𝑒 ∩ 𝐵 𝒚, 𝑒 = ∅, 𝒙, 𝒚 ∈ 𝐶, 𝒙 ≠ 𝒚

• and

𝐵 𝒙, 𝑒

𝒙∈𝐶

= 𝑈

• where 𝐵 𝒙, 𝑒 = 𝒘 ∶ 𝑑 𝒙,𝒘 ≤ 𝑒 .

• Perfect code is nontrivial if 𝐶 ≥ 2 and 𝑒 ≥ 1.

13


• In the following, we observe codes in Δ𝑛𝑞−1

under the metric

𝑑′ 𝒙, 𝒚 =1

2 𝑥𝑖 − 𝑦𝑖

𝑞−1

𝑖=0

– 𝒙 − 𝒚 1 is even for 𝒙, 𝒚 ∈ Δ𝑛𝑞−1

– This is equivalent to taking the “injection” distance 𝐷′

14


• Theorem:

15


• Theorem:

– Nontrivial 𝑒-perfect multiset code of length 𝑛 over a binary

alphabet exists for any 𝑛 ≥ 2𝑒 + 1. Such a code has 𝑛+1

2𝑒+1

codewords.

15


• Theorem:



2𝑒+1

codewords.

– Nontrivial 𝑒-perfect multiset code of length 𝑛 over a ternary alphabet exists if and only if 𝑛 = 3𝑒 + 1. Furthermore, there are exactly two 𝑒-perfect multiset codes of length 3𝑒 + 1, each having three codewords.

15


• Theorem:



2𝑒+1

codewords.

– Nontrivial 𝑒-perfect multiset code of length 𝑛 over a ternary alphabet exists if and only if 𝑛 = 3𝑒 + 1. Furthermore, there are exactly two 𝑒-perfect multiset codes of length 3𝑒 + 1, each having three codewords.

– Nontrivial 𝑒-perfect multiset codes over larger alphabets do not exist, for any 𝑒 and 𝑛.

15

Perfect multiset codes – Binary alphabet

16


• The space

Δ𝑛1 = 𝑛 − 𝑡, 𝑡 ∶ 𝑡 = 0,… , 𝑛

• can be represented as a graph with Δ𝑛1 = 𝑛 + 1 vertices, the “leftmost” vertex being (𝑛, 0), and the “rightmost” (0, 𝑛).

16


• The space

Δ𝑛1 = 𝑛 − 𝑡, 𝑡 ∶ 𝑡 = 0,… , 𝑛

• can be represented as a graph with Δ𝑛1 = 𝑛 + 1 vertices, the “leftmost” vertex being (𝑛, 0), and the “rightmost” (0, 𝑛).

• Figure: 1-perfect code in Δ81 (𝑒 = 1, 𝑛 = 8)

16


• Given any 𝑛 = 𝑘 2𝑒 + 1 + 𝑟 , with 𝑘 ≥ 1, 0 ≤ 𝑟 < 2𝑒 + 1, 𝑒-perfect code can be constructed as follows

𝐶 = 𝑛 −𝑟

2− 2𝑒 + 1 𝑖 ,

𝑟

2+ 2𝑒 + 1 𝑖 ∶ 𝑖 = 0,… , 𝑘

• This code has 𝑘 + 1 =𝑛+1

2𝑒+1 codewords.

17


• Given any 𝑛 = 𝑘 2𝑒 + 1 + 𝑟 , with 𝑘 ≥ 1, 0 ≤ 𝑟 < 2𝑒 + 1, 𝑒-perfect code can be constructed as follows

𝐶 = 𝑛 −𝑟

2− 2𝑒 + 1 𝑖 ,

𝑟

2+ 2𝑒 + 1 𝑖 ∶ 𝑖 = 0,… , 𝑘

• This code has 𝑘 + 1 =𝑛+1

2𝑒+1 codewords.

• Stronger statement: There are 𝑀 = min 𝑟 + 1 , 2𝑒 + 1 − 𝑟

perfect codes in Δ𝑛1 , and they can be enumerated.

17

Perfect multiset codes – Ternary alphabet

19


• The space

Δ𝑛2 = 𝑛 − 𝑡1 − 𝑡2, 𝑡1, 𝑡2 ∶ 𝑡1, 𝑡2 ≥ 0, 𝑡1 + 𝑡2 ≤ 𝑛

• can be represented as a triangular grid graph

19


• There are exactly two e-perfect codes in Δ3𝑒+12 :

𝐶1 = 2𝑒 + 1, 𝑒, 0 , 0, 2𝑒 + 1, 𝑒 , 𝑒, 0, 2𝑒 + 1

𝐶2 = 2𝑒 + 1, 0, 𝑒 , 𝑒, 2𝑒 + 1, 0 , 0, 𝑒, 2𝑒 + 1

20


• There are exactly two e-perfect codes in Δ3𝑒+12 :

𝐶1 = 2𝑒 + 1, 𝑒, 0 , 0, 2𝑒 + 1, 𝑒 , 𝑒, 0, 2𝑒 + 1

𝐶2 = 2𝑒 + 1, 0, 𝑒 , 𝑒, 2𝑒 + 1, 0 , 0, 𝑒, 2𝑒 + 1

• There are no e-perfect codes in Δ𝑛2 for 𝑛 ≠ 3𝑒 + 1.

20


• Proof: The proof consists of the following observations

21



– Vertex 𝑛, 0, 0 has to be covered by a ball around some codeword, and hence there must exist a codeword of the form 𝑛 − 𝑡, 𝑡1, 𝑡2 , with 𝑡1 + 𝑡2 = 𝑡 ≤ 𝑒

69



– Points of this form with 0 ≤ 𝑡 < 𝑒 or 0 < 𝑡1, 𝑡2 < 𝑡 cannot be codewords of an 𝑒-perfect code in Δ𝑛

2

22


• Proof: The proof consists of the following observations.

– The only remaining possibilities are 𝑛 − 𝑒, 𝑒, 0 and 𝑛 − 𝑒, 0, 𝑒

23




– When 𝑛 = 3𝑒 + 1 these two possibilities yield perfect codes 𝐶1 and 𝐶2

23




– When 𝑛 = 3𝑒 + 1 these two possibilities yield perfect codes 𝐶1 and 𝐶2

– When 𝑛 < 3𝑒 + 1 it is impossible to cover all points, and hence to build a perfect code

23



– There are no perfect codes for 𝑛 > 3𝑒 + 1

24

Perfect multiset codes – Larger alphabets

25


• There are no nontrivial perfect codes in Δ𝑛𝑞−1

for 𝑞 > 3.

25


• There are no nontrivial perfect codes in Δ𝑛𝑞−1

for 𝑞 > 3.

• The proof idea is the same: We try to build the perfect code and see that we can’t

– There is always a point that cannot be covered

25

Perfect codes in the Johnson space

26


• Johnson space 𝐽(𝑞, 𝑛) is the space of all binary sequences of length 𝑞 and Hamming weight 𝑛.

26


• Johnson space 𝐽(𝑞, 𝑛) is the space of all binary sequences of length 𝑞 and Hamming weight 𝑛. Clearly

𝐽 𝑞, 𝑛 ⊆ Δ𝑛𝑞−1

26




• Conjecture (Delsarte): There are no nontrivial perfect codes in 𝐽(𝑞, 𝑛).

26





• In our terminology: There are no nontrivial subset codes.

26





• In our terminology: There are no nontrivial subset codes.

• Can we use the same method to attack this problem?

26


• Observe 𝐽(9,5), and let 𝑒 = 1

27



• W.l.o.g. we can take the first codeword to be

𝒙 = 1 1 1 1 1 0 0 0 0

27




𝒙 = 1 1 1 1 1 0 0 0 0

• Observe the sequence

𝒗 = 1 1 1 0 0 1 1 0 0

• with 𝑑 𝒙, 𝒗 = 2.

27




𝒙 = 1 1 1 1 1 0 0 0 0

• Observe the sequence

𝒗 = 1 1 1 0 0 1 1 0 0

• with 𝑑 𝒙, 𝒗 = 2.

27

• W.l.o.g. we can assume that the codeword 𝒚 covering 𝒗 is

𝒚 = 1 1 0 0 0 1 1 1 0


• Now observe

𝒘 = 1 1 1 0 0 0 0 1 1

28


• Now observe

𝒘 = 1 1 1 0 0 0 0 1 1

• We have 𝑑 𝒙,𝒘 = 𝑑 𝒚,𝒘 = 2, so to cover 𝒘 we need another codeword 𝒛.

28


• Now observe

𝒘 = 1 1 1 0 0 0 0 1 1

• We have 𝑑 𝒙,𝒘 = 𝑑 𝒚,𝒘 = 2, so to cover 𝒘 we need another codeword 𝒛. But such a codeword (with 𝑑 𝒛,𝒘 = 1, 𝑑 𝒙, 𝒛 = 𝑑 𝒚, 𝒛 = 3) does not exist.

28


• Now observe

𝒘 = 1 1 1 0 0 0 0 1 1


• Therefore, 1-perfect code in 𝐽(9,5) does not exist.

28


• Now observe

𝒘 = 1 1 1 0 0 0 0 1 1


• Therefore, 1-perfect code in 𝐽(9,5) does not exist.

• More generaly, 𝑒-perfect in 𝐽(𝑞, 𝑛) cannot exist unless 𝑞 − 𝑛 ≡ 𝑛 ≡ 𝑒 (𝑚𝑜𝑑 𝑒 + 1) (Etzion)

28

Further…

29

Further…

• Constructions and decoding algorithms

29

Further…


• Bounds on multiset codes

29

Further…



• Multiset codes and multiset designs

29

Further…




• Do multiset codes have analogues in the vector space setting?

29

Further…




• Do multiset codes have analogues in the vector space setting?

– Can we define codes of arbitrary length and minimum distance for the operator channel?

29

coding for the permutation channel - universiteit gent

Documents