Skew cyclic codes: Hamming distance and decoding algorithms 1

Skew cyclic codes: Hamming distance and decoding algorithms 1 J. Gómez-Torrecillas, F. J. Lobillo, G. Navarro Department of Algebra and CITIC, University of Granada Department of Computer Sciences and AI, and CITIC, University of Granada NCRA V, Lens, June 14, 2017 1 Research supported by grants MTM2013-41992-P and MTM2016-78364-P from Spanish government and FEDER. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (1)

Based on: GLN, Generating idempotents in ideal codes, ACM Communications in Computer Algebra, 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 on Information Theory 62 (5) (2016), 2702 2706. GLN, A Sugiyama-like decoding algorithm for convolutional codes, 2016. arxiv:1607.07187. GLN, Ideal codes over separable ring extensions, IEEE Transactions on Information Theory 63 (5) (2017), 2796 2813. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (2)

Index 1 Motivation: Cyclic Convolutional Codes 2 Skew Reed-Solomon codes 3 Decoding GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (3)

Convolutional codes F a finite field, k n positive integers. A rate k/n convolutional transducer G transforms information sequences u =... u 1 u 0 u 1... (u i F k ) into code sequences v =... v 1 v 0 v 1... (v i F n ). Requirements: Write u = i= i 0 u i t i F k ((t)) = F((t)) k, v = j= j 0 v i t j F n ((t)) = F((t)) n. Then v = ug, where G is an k n full rank matrix with entries in F(t). GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (4)

Convolutional codes F a finite field, k n positive integers. A rate k/n convolutional transducer G transforms into information sequences u =... u 1 u 0 u 1... (u i F k ) code sequences v =... v 1 v 0 v 1... (v i F n ). Requirements: Write u = i= i 0 u i t i F k ((t)) = F((t)) k, v = j= j 0 v i t j F n ((t)) = F((t)) n. Then v = ug, where G is an k n full rank matrix with entries in F(t). Definition A rate k/n convolutional code D over F is the image of a rate k/n convolutional transducer G, that is D = {ug : u F k ((t))} F n ((t)). GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (4)

Cyclic convolutional codes (module version) Lemma The map D D F n [t] is a bijection between the set of rate k/n convolutional codes and the set of submodules of rank k of F n [t] = F[t] n that are direct summands. Definition (Module version) A rate k/n convolutional code is a rank k direct summand of F n [t]. Thus, a cyclic structure could be introduced by taking F n = A := F[x]/ x n 1, and suitable ideals of A[t] as cyclic convolutional codes. But... Problem: Piret (1976) observed that no non-block codes are obtained in this way. Solution: Piret s (1976) idea is skewing the multiplication of A[t]. Technically, this means to define cyclic convolutional codes as suitable left ideals of a skew polynomial ring A[t; σ]. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (5)

Cyclic convolutional codes (vector space version). Would we define a cyclic convolutional code as an ideal of F(t)[x]/ x n 1 = F(t) n? GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (6)

Cyclic convolutional codes (vector space version). Would we define a cyclic convolutional code as an ideal of F(t)[x]/ x n 1 = F(t) n? If so, then we get nothing but cyclic block codes (the factors of x n 1 have coefficients in F). Proposal: (GLN (2016)). A cyclic convolutional could be a left ideal of R = F(t)[x; σ]/ x n 1, where σ is an F automorphism of order n of F(t). Features: F(t)[x; σ] is a left and right Euclidean domain. F(t)[x; σ]/ x n 1 = M n (F(t) σ ). When the generator of the code is carefully chosen, we get an MDS code with respect to the Hamming distance. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (6)

Skew cyclic codes So, let L be a field, and σ : L L a field automorphism of finite order n. Set K = L σ. Consider the ring R = L[x; σ]/ x n 1 = M n (K). Note: The multiplication rule in L[x; σ] is xa = σ(a)x for all a L. We have the isomorphism of L vector spaces p : L n R sending (c 0, c 1,..., c n 1 ) onto c 0 + c 1 x + + c n 1 x n 1. Note: The classical identification of equivalence classes and its representatives is (and will be) made. Definition A k dimensional L linear code C L n of dimension n is said to be a skew cyclic code if p(c) is a left ideal of R. Note: We will identify C with p(c). Since every left ideal of R is principal, we will often speak of the skew code generated by a polynomial, aggravating the abuse of language. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (7)

Skew RS codes Idea: Choosing carefully generator polynomials of skew codes, we get very concrete results. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (8)

Skew RS codes Idea: Choosing carefully generator polynomials of skew codes, we get very concrete results. Construction method of Skew RS codes Choose a normal basis {α, σ(α),..., σ n 1 (α)} of the field extension L/K. Set β = α 1 σ(α) x n 1 decomposes as the least common left multiple x n 1 = [ x β, x σ(β), x σ 2 (β),..., x σ n 1 (β) ] l For 1 k < n, the left ideal C of R generated by g = [ x β, x σ(β), x σ 2 (β),..., x σ n k 1 (β) ] l is an SCC of length n and dimension k. We call them Reed-Solomon skew codes. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (8)

Parity check matrix A parity check matrix of the skew RS code is given by N 0 (β) N 0 (σ(β))... N 0 (σ n k 1 (β)) N 1 (β) N 1 (σ(β))... N 1 (σ n k 1 (β)) H = N 2 (β) N 2 (σ(β))... N 2 (σ n k 1 (β))....... N n 1 (β) N n 1 (σ(β))... N n 1 (σ n k 1 (β)) Here, for γ L, N j (γ) = γσ(γ)... σ j 1 (γ) is the remainder of the left division of x j by x γ, that is x j = q(x)(x γ) + N j (γ). GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (9)

Transmission and syndromes Our skew RS code C is generated by g = [ x β, x σ(β), x σ 2 (β),..., x σ n k 1 (β) ] l GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (10)

Transmission and syndromes Our skew RS code C is generated by g = [ x β, x σ(β), x σ 2 (β),..., x σ n k 1 (β) ] l Set τ = δ 1 2 = n k 2 which is the maximum number of errors that the code can correct. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (10)

Transmission and syndromes Our skew RS code C is generated by g = [ x β, x σ(β), x σ 2 (β),..., x σ n k 1 (β) ] l Set τ = δ 1 2 = n k 2 which is the maximum number of errors that the code can correct. m c = mg y = c + e k 1 m i x i i=0 n 1 c i x i i=0 n 1 ν c i x i + e j x kj i=0 j=1 ν τ message encoded received GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (10)

Error locator and error evaluator We thus know the syndrome polynomial, defined as S = 2τ 1 i=0 σ i (α)s i x i. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (11)

Error locator and error evaluator We thus know the syndrome polynomial, defined as Define the locator polynomial S = 2τ 1 i=0 σ i (α)s i x i. λ = [1 σ k1 (β)x, 1 σ k2 (β)x,..., 1 σ kν (β)x] r GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (11)

Error locator and error evaluator We thus know the syndrome polynomial, defined as Define the locator polynomial S = 2τ 1 i=0 σ i (α)s i x i. λ = [1 σ k1 (β)x, 1 σ k2 (β)x,..., 1 σ kν (β)x] r For 1 j ν we thus know that there is p j of degree ν 1 such that λ = (1 σ kj (β)x)p j. Define the evaluator polynomial ν ω = e j σ kj (α)p j (1) j=1 Sugiyama s decoding scheme. If λ is computed, then the error positions k 1,..., k ν are derived. With these at hand, we can compute p 1,..., p ν. Finally, the error values e 1,..., e ν are computed by solving a linear system from (1), whenever ω is known. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (11)

The non-commutative key equation Theorem These polynomials satisfy the non-commutative key equation for some u of degree smaller than ν. ω = Sλ + x 2τ u, GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (12)

The non-commutative key equation Theorem These polynomials satisfy the non-commutative key equation for some u of degree smaller than ν. Theorem The non-commutative key equation is a right multiple of the equation ω = Sλ + x 2τ u, x 2τ u + Sλ = ω (2) x 2τ u I + Sv I = r I, (3) where u I, v I and r I are the Bezout coefficients returned by the REEA with input x 2τ and S, and I is the index determined by the conditions deg r I 1 τ and deg r I < τ. In particular, λ = v I g and ω = r I g for some g L[x; σ]. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (12)

Sugiyama-like decoding If (λ, ω) r = 1, then the last theorem gives that the REEA serves for the computation of the locator polynomial λ and the evaluator polynomial ω. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (13)

Sugiyama-like decoding If (λ, ω) r = 1, then the last theorem gives that the REEA serves for the computation of the locator polynomial λ and the evaluator polynomial ω. As mentioned above, once these polynomials are computed, the error polynomial is computed by a scheme similar to the Sugiyama algorithm for (commutative) RS codes. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (13)

Sugiyama-like decoding algorithm Input: A polynomial y = n 1 i=0 y ix i received from the transmission of a codeword c in a skew RS code C generated by g = [{x σ i (β)} i=0,...,n k 1 ] l of error-correcting capacity τ = n k 2. Output: A codeword c, or key equation failure. 1: for 0 i 2τ 1 do 2: S i n 1 j=0 y jn j (σ i (β)) 3: S 2τ 1 i=0 σi (α)s i x i 4: If S = 0 then 5: Return y 6: {u i, v i, r i } i=0,...,l REEA(x 2τ, S) 7: I first iteration in REEA with deg r i < τ, pos 8: for 0 i n 1 do 9: If 1 σ i (β)x is a left divisor of v I then 10: pos = pos {i} 11: If deg v I > Cardinal(pos) then 12: Return key equation failure 13: for j pos do 14: p j right-quotient(v I, 1 σ j (β)x) 15: Solve the linear system r I = j pos e jσ j (α)p j 16: e j pos e jx j 17: Return y e GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (14)

Key equation failures How often occurs the key equation failure? GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (15)

Key equation failures How often occurs the key equation failure? Well, it never happens if the error values are in general position. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (15)

Key equation failures How often occurs the key equation failure? Well, it never happens if the error values are in general position. More precisely, Theorem The key equation failure happens if and only if e 1,..., e ν are linearly dependent over K = L σ. GLN (UGR) Skew RS codes NCRA V, June 14, 2017 (15)