On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots

We determine the structure of cyclic codes over Z 4 for arbitrary even length giving the generator polynomial for these codes. We determine the number of cyclic codes for a given length. We describe the duals of the cyclic codes, describe the form of cyclic codes that are self-dual and give the number of these codes. We end by examining specific cases of cyclic codes, giving all cyclic self-dual codes of length less than or equal to 14.

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“…In fact, this identification is the same as the one given in [5], Section 3 (the φ there is our −1 ).…”

Section: Dualsmentioning

confidence: 90%

Cyclic Codes Over $$\mathbb{Z}_{4}$$ of Even Length

Dougherty

Ling

2006

Des Codes Crypt

Self Cite

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“…In this situation, it is possible to decompose C in a similar way, but the projections are codes defined over finite chain rings. The reader can refer to [7] for more details.…”

Section: Corollary 2 Suppose That F (X) Satisfies the Conditions Of mentioning

confidence: 99%

Codes over $\mathcal{L}(GF(2)^m,GF(2)^m)$, MDS Diffusion Matrices and Cryptographic Applications

Berger

Amrani

2015

Lecture Notes in Computer Science

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Abstract. The aim of this paper is to provide a general framework in the study of binary block codes. The main objective is to present a general approach in order to explore MDS diffusion matrices used for example in the design of block ciphers with a Substitution Permutation Network design (the so-called SPN block-ciphers).In order to analyze these codes, we consider additive block codes over binary m-tuples. We are interested in the distance properties related to the block structure. To do this, we introduce a notion of L-codes that are codes over the non-commutative ring of linear endomorphisms of GF (2) m . We study the main properties of these codes, especially the notion of duality in this context. We show how most of the known families of block codes can be interpreted in this context. Finally, we conclude by practical examples that allow to derive MDS diffusion matrices over GF (2) m from MDS matrices constructed over smaller blocks.

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“…Quasi-cyclic codes (see [8,[11][12][13][14][15]20]) are defined as linear block codes of length N over a finite field F q which are invariant with respect to a cyclic shift by n positions for some integer n, 0 < n ≤ N . When n = 1, they correspond to the classical cyclic codes.…”

Section: A Class Of Quasi-cyclic Codes Over Finite Fieldsmentioning

confidence: 99%

On self-orthogonal group ring codes

Feng

2008

Des. Codes Cryptogr.

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We obtain structural results about group ring codes over F [G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F q [u], u r = 0 with r > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.

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On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots

Cited by 65 publications

References 18 publications

Cyclic Codes Over $$\mathbb{Z}_{4}$$ of Even Length

Cyclic Codes Over $$\mathbb{Z}_{4}$$ of Even Length

Codes over $\mathcal{L}(GF(2)^m,GF(2)^m)$, MDS Diffusion Matrices and Cryptographic Applications

On self-orthogonal group ring codes

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