On the subfield subcodes of Hermitian codes

We present a fast algorithm using Gröbner basis to compute the dimensions of subfield subcodes of Hermitian codes. With these algorithms we are able to compute the exact values of the dimension of all subfield subcodes up to q ≤  32 and length up to 2¹⁵. We show that some of the subfield subcodes of Hermitian codes are at least as good as the previously known codes, and we show the existence of good long codes.     

Galois invariance,trace codes and subfield subcodes

Given a Galois extension we relate subfield subcodes with trace codes showing that a code is invariant under the Galois group if and only if its restriction coincides with the trace code.     

Asymptotically good families of subfield subcodes of geometric Goppa codes

We show that subfield subcodes of certain geometric Goppa codes meet the Gilbert-Varshamov bound. A very special case of our theorem is the well known fact that classical Goppa codes meet the Gilbert-Varshamov bound.     

Linear recurring sequences over modules

The aim of this paper is to extend some fundamental and applied results of the theory of linear recurring sequences over fields to the case of polylinear recurring sequences over rings and modules. Quasi-Frobenius modules and Galois rings play a very special role in this project.     

Linear recurring sequences over rings and modules

To the 80th anniversary of the birth of Alexander Illarionovich Uzkow (1913–1990)     

Linear recurring sequences over Galois rings

Linear recurrences of maximal period over a Galois ring and over a residue class ring modulo p are studied. For any such recurrence, the coordinate sequences (in p-adic and some other expansions) are considered as linear recurring sequences over a finite field. Upper and lower bounds for the ranks (linear complexities) of these coordinate sequences are obtained. The results are based on using the properties of Galois rings and the trace-function on such rings.Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 169–189, March-April, 1995.     

Symmetry codes and their invariant subcodes

We define and study the invariant subcodes of the symmetry codes in order to be able to determine the algebraic properties of these codes. An infinite family of self-orthogonal rate $\text{1}\text{2}$ codes over GF(3), called symmetry codes, were constructed in [3]. A (2q + 2, q + 1) symmetry code, denoted by C(q), exists whenever q is an odd prime power ≡ ?1, (mod 3). The group of monomial transformations leaving a symmetry code invariant is denoted by G(q). In this paper we construct two subcodes of C(q) denoted by R_σ(q) and R_μ(q). Every vector in R_σ(q) is invariant under a monomial transformation τ in G(q) of odd order s where s divides (q + 1). Also R_μ(q) is invariant under τ but not vector-wise. The dimensions of R_σ(q) and R_μ(q) are determined and relations between these subcodes are given. An isomorphism is constructed between R_σ(q) and a subspace of $\text{W = V}{}_3{}^{\mathrm{<mtext>(2q+2)</mtext><mtext>s</mtext>}}$. It is shown that the image of R_σ(q) is a self-orthogonal subspace of W. The isomorphic images of R_σ(17) (under an order 3 monomial) and R_σ(29) (under an order 5 monomial) are both demonstrated to be equivalent to the (12, 6) Golay code.     

Subspace subcodes of generalized reed-solomon codes

1. IntroductionTlle subfield subcodes of RS Cud('s a11(l tl1" tra('e-sIlurtened RS codes are introducedill [1] al1d [2], respe(!tiv'lly. II1 [3] the subsI)acc subcodes ()f RS codes over GF(2"') areintrodllced, which caIl be rcgarded as a ge11eralizatio11 of the subfield subcodes of RS codesa11d tlle trace-shorte1led RS codes, a11d tl1c fOrll1ulas t() ('oIllI)llte tl1e diInensions of thesubsPace subc(,des ()f RS c()des ()ver GF(2"' ) are given. llut what we call get if, i11stead ()fR…     

Linear codes from support designs of ternary cyclic codes

Designs, Codes and Cryptography - For a long time, the literature has demonstrated that designs and codes are exciting topics for combinatorics and coding theory. Linear codes and t-designs are, in...     

Intersection of Hamming codes avoiding Hamming subcodes

We prove that given a binary Hamming code \({{\mathcal{H}}^n}\) of length n = 2 ^m ? 1, m ≥ 3, or equivalently a projective geometry PG(m ? 1, 2), there exist permutations \({\pi \in \mathcal{S}_n}\) , such that \({{\mathcal{H}}^n}\) and \({\pi({\mathcal{H}}^n)}\) do not have any Hamming subcode with the same support, or equivalently the corresponding projective geometries do not have any common flat. The introduced permutations are called AF permutations. We study some properties of these permutations and their relation with the well known APN functions.     

Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Designs, Codes and Cryptography - In this paper, we construct a family of non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Moreover, we present a...     

Construction of deletion correcting codes using generalized Reed–Solomon codes and their subcodes

A code is n-deletion correcting if it is possible to correct any n deletion of symbols having occurred in transmission of codewords. In this paper, we present explicit constructions of n-deletion correcting codes for arbitrary values of n using generalized Reed–Solomon codes and their subcodes.      

The relative generalized Hamming weight of linear q-ary codes and their subcodes

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.      

On the reconstruction of binary codes from the dimensions of their subcodes

We prove that to reconstruct an arbitrary binary code up to equivalence it suffices to know the dimensions of all subcodes of even cardinality. It is shown that this theorem cannot be improved.     

Linear error-block codes

A linear error-block code is a natural generalization of the classical error-correcting code and has applications in experimental design, high-dimensional numerical integration and cryptography. This article formulates the concept of a linear error-block code and derives basic results for this kind of code by direct analogy to the classical case. Some problems for further research are raised.     

Hermitian LCD codes from cyclic codes

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. It was proved that asymptotically good Hermitian LCD codes exist. The objective of this paper is to construct some cyclic Hermitian LCD codes over finite fields and analyse their parameters. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented. Most Hermitian LCD codes presented in this paper are not BCH codes.     

Quasi-cyclic codes with cyclic constituent codes

Ling and Solé [S. Ling, P. Solé, On the algebraic structure of quasi-cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47 (2001) 2751–2760] showed that every quasi-cyclic code C is constructed from shorter linear codes which are called the constituent codes of C. Given a quasi-cyclic code C of length ℓm and index ℓ with m being pairwise coprime to ℓ and the order of the field C is over, if all its constituent codes are cyclic with their zeroes having full multiplicity, C is then shown to be equivalent to a cyclic code whose zeroes with their multiplicities are fully described in terms of the nonzeroes of the cyclic constituent codes. The general transformation to obtain the above-mentioned equivalent cyclic code is also explicitly given. The approach adopted here follows the approach used by A.M.A. Natividad [A.M.A. Natividad, PhD thesis, Department of Mathematics, University of Philippines Diliman, The Philippines, 2004] and uses the generalized discrete Fourier transform on the algebraic structure of the class of quasi-cyclic codes developed by Ling and Solé [S. Ling, P. Solé, On the algebraic structure of quasi-cyclic codes I: Finite fields, IEEE Trans. Inform. Theory 47 (2001) 2751–2760].