A method to construct girth-12 (3,L) quasi-cyclic lowdensity parity-check (QC-LDPC) codes with all lengths larger than a certain given number is proposed, via a given girth-12 code subjected to some constraints. The lengths of these codes can be arbitrary integers of the form LP, provided that P is larger than a tight lower bound determined by the maximal element within the exponent matrix of t...

A length $ml$, index $l$ quasi-cyclic code can be viewed as a cyclic of $m$ over the field $\mathbb F_{q^l}$ via basis extension F_{q^l}/\mathbb F_{q}$. This is an additive code. In [C. Güneri, F. Özdemir, P. Solé, On structure codes, Discrete. Math., 341 (2018), 2735-2741], authors characterize $(l,m)$ values for one-generator codes which it impossible to have F_{q^l}$-linear image any choice ...

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4m ≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 237, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers...

We study covering codes of permutations with the l∞-metric. We provide a general code construction, which uses smaller building-block codes. We study cyclic transitive groups as building blocks, determining their exact covering radius, and showing linear-time algorithms for finding a covering codeword. We also bound the covering radius of relabeled cyclic transitive groups under conjugation. In...

At the ECRYPT Hash Workshop 2007, Finiasz, Gaborit, and Sendrier proposed an improved version of a previous provably secure syndrome-based hash function. The main innovation of the new design is the use of a quasi-cyclic code in order to have a shorter description and to lower the memory usage. In this paper, we look at the security implications of using a quasi-cyclic code. We show that this v...

Conditions on the existence of self-dual θ-codes defined over a finite field IFq are studied for θ automorphism of IFq. When q ≡ 1 (mod 4) it is proven that there always exists a self-dual θ-code in any dimension and that self-dual θ-codes of a given dimension are either all θ-cyclic or all θ-negacyclic. When q ≡ 3 (mod 4), there does not exist a selfdual θ-cyclic code and a necessary and suffi...

At the ECRYPT Hash Workshop 2007, Finiasz, Gaborit, and Sendrier proposed an improved version of a previous provably secure syndrome-based hash function. The main innovation of the new design is the use of a quasi-cyclic code in order to have a shorter description and to lower the memory usage. In this paper, we look at the security implications of using a quasi-cyclic code. We show that this v...

We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret and disguise it by considering its shortened expanded code the base field. Considering provides safeguard against distinguisher attacks based on Schur product. Moreover, without using cyclic or quasi-cyclic structure obtain key size reduction of near...

If a subcode of the second order Reed-Mul ler code of length 2 ~ has minimum distance 2 ~ ~ 2 (1/2)~-~ then it has at most 2 2~ words. A generalized Kerdock code is defined to be such a subcode in which this max imum is attained. Such codes were first constructed by Kerdock [7]. His codes are extended cyclic codes, in the sense that there is an au tomorphism of order 2" 1 fixing one coordinate ...