j.gómez-torrecillas ,f.j.lobillo , y gabrielnavarrozsta/pres/navarro.pdf · skew cyclic codes1...

Skew cyclic codes1

J. Gómez-Torrecillas†, F. J. Lobillo†, Gabriel Navarro‡

†Department of Algebra and CITIC, University of Granada‡Department of Computer Sciences and AI, University of Granada

STA’17, Prague, September 6th, 2017

1Supported by grants MTM2013-41992-P and MTM2016-78364-P from Spanish government and FEDER.GLN (UGR) Skew cyclic codes STA’17 (1)

Based on:GLN, Generating idempotents in ideal codes, ACM Communications in ComputerAlgebra, Vol 48, No. 3, Issue 189, September 2014, ISSAC poster abstracts, pp. 113-115.GLN, A new perspective of cyclicity in convolutional codes, IEEE Transactions onInformation Theory 62 (5) (2016), 2702–2706.GLN, Peterson-Gorenstein-Zierler algorithm for skew RS codes, Linear andMultilinear Algebra, 2017.GLN, A Sugiyama-like decoding algorithm for convolutional codes, IEEETransactions on Information Theory 2017.GLN, Ideal codes over separable ring extensions, IEEE Transactions on InformationTheory 63 (5) (2017), 2796–2813.

GLN (UGR) Skew cyclic codes STA’17 (2)

https://doi.org/10.1109/TIT.2016.2538264

http://dx.doi.org/10.1080/03081087.2017.1301364

https://doi.org/10.1109/TIT.2017.2731774

Index

1 Motivation: Coding Theory and cyclic codes

2 Cyclicity in convolutional codes

3 Skew cyclic codes


Index



3 Skew cyclic codes


Scheme of encoding-decoding

A code is simply a set

But it is quite common to consider vector spaces (linear codes) and linear maps


Linear codes (Golay 1949, Hamming 1950, ...)

0→ Fk G−→ Fn H−→ Fn−k → 0

The code is G(Fk) (length n, dimension k)

Advantages of Linear Algebra:Encoding is easy, Gu = v (transforms k-symbols into n-symbols)Post-decoding is easy, G′G = I so G′v = u (transform n-symbols into k-symbols)Check errors is easy, Hv = 0 (right), Hv 6= 0 (errors)Correcting is easy (but not efficient)

received=codeword+error→ H(received) = H(codeword)︸︷︷︸0

+H(error)


Cyclic block codes (Bose-Haudhuri-Hocquenghem 1959, Reed-Solomon 1960,...)

⇐⇒ a code is an ideal ofF[x]〈xn − 1〉

Fn p−→F[x]〈xn − 1〉

(a0, a1, . . . , an−1) 7−→ a0 + a1x+ · · ·+ an−1xn−1

We may work in an Euclidean Domain, semisimple (charF - n),...


Cyclic block codes (Bose-Haudhuri-Hocquenghem 1959, Reed-Solomon 1960,...)

〈f〉 ≤ F[x]〈xn − 1〉

, deg f = n− k

Encoding is easy, v = uf

Post-decoding is easy, u = quotient(v, f)

Check errors is easy, f | v (right), f - v (errors)Correcting is not so easy (but more efficient)

I Peterson-Gorenstein-Zierler (Linear Algebra)I Sugiyama (Euclidean Algorithm)I Berlekamp-Massey (Iterative)


Convolutional codes (Forney, 1970)An (n, k,m) convolutional encoder will encode a k-bit input block into an n-bit ouput block, whichdepends on the current input block and the m preceding input blocks

vn = G0un +G1un−1 + · · ·+Gmun−m

Delay operatorIf t represents the delay operator, that is, un−1 = tun, then we get

vn = (G0 +G1t+ · · ·+Gmtm)un

Thus, the encoder G = G0 +G1t+ · · ·+Gmtm ∈Mk×n(F[t])


Convolutional codes (Forney, 1970)

AlgebraicallyA convolutional (linear) code is a F[t]–direct summand of F[t]n (generated by the rows of G)

0→ F[t]k G−→ F[t]n H−→ F[t]n−k → 0

Linear Algebra over a commutative PID!

Technical remark: why a direct summand?We want to post-decode, i.e, an left inverse matrix G′

G′ should be polynomial

Then, G has Smith canonical form(Ik 00 0

)The image of G (the code) is a direct summand


Convolutional codes (Forney, 1970)

AlgebraicallyA convolutional (linear) code is a F[t]–direct summand of F[t]n (generated by the rows of G)

0→ F[t]k G−→ F[t]n H−→ F[t]n−k → 0

Linear Algebra over a commutative PID!

Encoding, post-decoding, check errors as linear block codesCorrecting...Viterbi Algorithm, pretty much industrial standard!

Dynamic ProgrammingMaximum-Likehood decodingExponential in memory/base field


Index



3 Skew cyclic codes


Cyclic convolutional codes (Piret 1975)First Attempt

⇐⇒ideal of

F[t][x]〈xn − 1〉+

direct summand of F[t]n

F[t]n p−→F[t][x]〈xn − 1〉

(p0, p2, . . . , pn−1) 7−→ p0 + p1x+ · · ·+ pn−1xn−1


Cyclic convolutional codes (Piret 1975)First Attempt

⇐⇒ideal of

F[t][x]〈xn − 1〉+

direct summand of F[t]n

F[t][x]〈xn − 1〉

∼= 〈π1〉 ⊕ · · · ⊕ 〈πs〉 ⊕ 〈πs+1〉 ⊕ · · · ⊕ 〈πr〉︸︷︷︸C=〈f〉,f∈F[x]

too commutative!

It is a cyclic block code


Cyclic convolutional codes (Piret 1976, Ross 1979, Gluesing-Luerssen and Schmale 2004)

Second Attempt

Piret: remove commutativity of the working-ambient algebra

F[t]-modules︷︸︸︷F[t]n ∼= Fn[t] ∼=

F[x]〈xn − 1〉

[t]

↑ ↑∼=

(∏i

Fi

)[t]

vector ofpolynomials

polynomialof vectors

A solution: skew multiplication yieldingF[x]〈xn − 1〉

[t;σ]

Algebraically

A σ-CCC is a left ideal ofF[x]〈xn − 1〉

[t;σ] + direct summand of F[t]n


Cyclic convolutional codes (Piret 1976, Ross 1979, Gluesing-Luerssen and Schmale 2004)

Second Attempt

Advantages of this perspective (Gluesing-Luerssen and Schmale 2004)There exist non-block cyclic convolutional codesExtend block cyclicityCCCs are principalGenerated by an idempotent


Ideal codes (López-Permouth and Szabo 2013)Second Attempt

Extend the word-ambient algebraF[x]〈xn − 1〉

to other F-algebras

Ideal codeGiven a n-dimensional word-ambient F-algebra A and a sentence-ambient

A[t;σ, δ]v−→∼= F[t]n,

an ideal code I is a direct summand of F[t]n such that v−1(I) is a left ideal of A[t;σ, δ]

Finite semisimple group algebras FG (Estrada et al. 2008)... and commutative semisimple algebras (López-Permouth and Szabo 2013)Matrix algebrasMn(G) (GLN 2015)Separable extensions F ⊂ A (GLN 2017)


Ideal codes (GLN 2017)Second Attempt

Problem 1Are direct summands as left ideals? (Split ideal code)

An answer (GLN 2017)Whenever F ⊂ A has a separability element p and (σ ⊗ σ)(p) = p

(⇒ F[t] ⊂ A[t;σ] separable ⇒ F[t]-direct summands are A[t;σ]-direct summands)

Then, there is a generating idempotent...and an algorithm for explicit computation


Ideal codesSecond Attempt

Problems of this perspectiveA[t;σ] is not a PID → many technical difficultiesNo Euclidean division, gcd, lcm, ...Decoding using cyclic structures is difficultComputing distances is difficult


Index



3 Skew cyclic codes


Convolutional codes (vector space version)

Original point-of-viewConsider the extension of scalars

MF[t]Q=−⊗F[t]F(t) //MF(t)

{direct summands of F[t]n} ooQ // {vector subspaces of F(t)n}

Definition (vector space version)A rate k/n convolutional code is a k-dimensional vector subspace of F(t)n.


Skew cyclic convolutional codes (GLN, 2016)Third Attempt

Would we define a cyclic convolutional code as an ideal of F(t)[x]/〈xn − 1〉 ∼= F(t)n?If so, then we get nothing but cyclic block codes (the factors of xn − 1 have coefficients in F).

GLN 2016A skew cyclic convolutional code could be a left ideal of R = F(t)[x;σ]/〈xn − 1〉, where σ is anF–automorphism of order n of F(t).

Features:

◦ F(t)[x;σ] is a left and right Euclidean domain.

◦ F(t)[x;σ]/〈xn − 1〉 ∼=Mn(F(t)σ).

◦ When the generator of the code is carefully chosen, we get an MDS code with respect to theHamming distance.

◦ The theory (including the algorithms) work for any field L.


Skew cyclic codesThird Attempt

GLN 2017A skew cyclic code over a field L is a left ideal of L[x;σ]/〈xn − 1〉, where σ is an automorphism oforder n of L.

L = F, skew cyclic block codesL = F(t), skew cyclic convolutional codesL = Q(χ), skew cyclic cyclotomic codes (?)...


Skew cyclic codes

Systematic constructionFind α (non-constant) with {α, σ(α), . . . , σn−1(α)} basis of Lσ ⊂ L

I Normal Basis Theorem...or,I Random search

Set β = α−1σ(α), [x− β, x− σ(β), . . . , x− σn−1(β)

]`= xn − 1

Choose {i1, i2, . . . , ik} ⊂ {0, 1, . . . , n− 1},

g =[x− σi1(β), x− σi2(β), . . . , x− σik(β)

]`

〈g〉 is a sCC of dimension n and dimension n− k


Skew Reed-Solomon codes (GLN, 2017)

Skew RS codes)A skew RS code of dimension k = n− δ + 1 is a sCC generated by[

x− β, x− σ(β), . . . , x− σδ−2(β)]`

AdvantagesIt has minimum distance δ (MDS)Decoding algorithms avaliable

I Sugiyama-like algorithm (GLN, 2017)I Peterson-Gorenstein-Zierler algorithm (GLN, 2017)


j.gómez-torrecillas ,f.j.lobillo , y gabrielnavarrozsta/pres/navarro.pdf · skew cyclic codes1...

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