Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

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].     

Cover-free codes and separating system codes

A set \(T\subset {GF(q)}\), \(q=p^h\) is a super-Vandermonde set if \(\sum _{y\in T} y^k=0\) for \(0< k <|T|\). We determine the structure of super-Vandermonde sets of size \(p+1\) (almost small) and size \(q/p-1\) (almost large).     

Hermitian codes as generalized Reed-Solomon codes

Hermitian codes obtained from Hermitian curves are shown to be concatenated generalized Reed-Solomon codes. This interpretation of Hermitian codes is used to investigate their structure. An efficient encoding algorithm is given for Hermitian codes. A new general decoding algorithm is given and applied to Hermitian codes to give a decoding algorithm capable of decoding up to the full error correcting capability of the code.This work is supported by a Natural Science and Engineering Research Council Grant A7382.     

Self-dual 2-quasi-cyclic codes and dihedral codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if $\mathrm{char}F=2$), or a consta-dihedral code (if $\mathrm{char}F\ne 2$). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other.     

AMDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13.     

New binary codes from extended Goppa codes

An efficient construction of extended length Goppa codes is presented. The construction yields four new binary codes [153, 71, 25], [151, 70, 25], [160, 70, 27], and [158, 69, 27]. The minimum distances are larger than those of the best previously known linear codes of the same length and dimension.     

MDS symbol-pair codes from repeated-root cyclic codes

Designs, Codes and Cryptography - Symbol-pair codes are proposed to protect against pair-errors in symbol-pair read channels. The research of symbol-pair codes with the largest possible minimum...     

Binary permutation sequences as subsets of Levenshtein codes,spectral null codes,run-length limited codes and constant weight codes

We investigate binary sequences which can be obtained by concatenating the columns of (0,1)-matrices derived from permutation sequences. We then prove that these binary sequences are subsets of a surprisingly diverse ensemble of codes, namely the Levenshtein codes, capable of correcting insertion/deletion errors; spectral null codes, with spectral nulls at certain frequencies; as well as being subsets of run-length limited codes, Nyquist null codes and constant weight codes. This paper was presented in part at the IEEE Information Theory Workshop, Chengdu, China, October, 2006.     

New entanglement-assisted quantum codes from k-Galois dual codes

Entanglement-assisted quantum error-correcting (EAQEC, for short) codes use pre-existing entanglements between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, they can only be constructed from classical linear codes which contain their Hermitian dual codes. However, how to determine the parameters of ebits c in EAQEC codes is not an easy task. In this paper, let p be prime and e, k be integers, we construct six classes of EAQEC codes based on k-Galois dual codes over finite fields ${\mathbb{F}}_{p^e}$, where $0\le k<e$. The parameter of ebits c of these EAQEC codes can be easily generated algebraically. Furthermore, the six classes of EAQEC codes are of maximal entanglement, most of which have better parameters than current EAQEC codes available.     

New quantum codes from evaluation and matrix-product codes

Stabilizer codes obtained via the CSS code construction and the Steane's enlargement of subfield-subcodes and matrix-product codes coming from generalized Reed–Muller, hyperbolic and affine variety codes are studied. Stabilizer codes with good quantum parameters are supplied; in particular, some binary codes of lengths 127 and 128 improve the parameters of the codes in http://www.codetables.de. Moreover, non-binary codes are presented either with parameters better than or equal to the quantum codes obtained from BCH codes by La Guardia or with lengths that cannot be reached by them.     

Quasi-cyclic codes as codes over rings of matrices

Quasi-cyclic codes over a finite field are viewed as cyclic codes over a noncommutative ring of matrices over a finite field. This point of view permits to generalize some known results about linear recurring sequences and to propose a new construction of some quasi-cyclic codes and self-dual codes.