Near MDS poset codes and distributions

We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the "ordered metric" characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.Comment: 13 pages, 1 figur

Error-block codes and poset metrics

Let P = ({1, 2,..., n}, <=) be a poset, let V-1, V-2,...,V-n, be a family of finite-dimensional spaces over a finite field F-q and let V = V-1 circle plus V-2 circle plus ... V-n. In this paper we endow V with a poset metric such that the P-weight is constant on the non-null vectors of a component V-i, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code [8; 4; 4] to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code [24; 12; 8] into perfect codes. We also give a complete description of the groups of linear isometrics of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometrics of the error-block metric spaces.Let P = ({1, 2,..., n}, <=) be a poset, let V-1, V-2,...,V-n, be a family of finite-dimensional spaces over a finite field F-q and let V = V-1 circle plus V-2 circle plus ... V-n. In this paper we endow V with a poset metric such that the P-weight is cons2195111FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOsem informaçã

Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition coincides the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures

Constacyclic codes of length 4ps4p^s over the Galois ring GR(pa,m)GR(p^a,m)

For prime $p$, $GR(p^a,m)$ represents the Galois ring of order $p^{am}$ and characterise $p$, where $a$ is any positive integer. In this article, we study the Type (1) $\lambda$-constacyclic codes of length $4p^s$ over the ring $GR(p^a,m)$, where $\lambda=\xi_0+p\xi_1+p^2z$, $\xi_0,\xi_1\in T(p,m)$ are nonzero elements and $z\in GR(p^a,m)$. In first case, when $\lambda$ is a square, we show that any ideal of $\mathcal{R}_p(a,m,\lambda)=\frac{GR(p^a,m)[x]}{\langle x^{4p^s}-\lambda\rangle}$ is the direct sum of the ideals of $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}-\delta\rangle}$ and $\frac{GR(p^a,m)[x]}{\langle x^{2p^s}+\delta\rangle}$. In second, when $\lambda$ is not a square, we show that $\mathcal{R}_p(a,m,\lambda)$ is a chain ring whose ideals are $\langle (x^4-\alpha)^i\rangle\subseteq \mathcal{R}_p(a,m,\lambda)$, for $0\leq i\leq ap^s$ where $\alpha^{p^s}=\xi_0$. Also, we prove the dual of the above code is $\langle (x^4-\alpha^{-1})^{ap^s-i}\rangle\subseteq \mathcal{R}_p(a,m,\lambda^{-1})$ and present the necessary and sufficient condition for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1) $\lambda$-constacyclic codes of length $4p^s$ are obtained when $\lambda$ is not a square.Comment: This article has 18 pages and ready to submit in a journa

Equivalence Theorems and the Local-Global Property

In this thesis we revisit some classical results about the MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight-preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study a local-global property that naturally generalizes the MacWilliams equivalence theorems. Making use of F-partitions, we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid

Bounds on the size of codes

In this dissertation we determine new bounds and properties of codes in three different finite metric spaces, namely the ordered Hamming space, the binary Hamming space, and the Johnson space. The ordered Hamming space is a generalization of the Hamming space that arises in several different problems of coding theory and numerical integration. Structural properties of this space are well described in the framework of Delsarte's theory of association schemes. Relying on this theory, we perform a detailed study of polynomials related to the ordered Hamming space and derive new asymptotic bounds on the size of codes in this space which improve upon the estimates known earlier. A related project concerns linear codes in the ordered Hamming space. We define and analyze a class of near-optimal codes, called near-Maximum Distance Separable codes. We determine the weight distribution and provide constructions of such codes. Codes in the ordered Hamming space are dual to a certain type of point distributions in the unit cube used in numerical integration. We show that near-Maximum Distance Separable codes are equivalently represented as certain near-optimal point distributions. In the third part of our study we derive a new upper bound on the size of a family of subsets of a finite set with restricted pairwise intersections, which improves upon the well-known Frankl-Wilson upper bound. The new bound is obtained by analyzing a refinement of the association scheme of the Hamming space (the Terwilliger algebra) and intertwining functions of the symmetric group. Finally, in the fourth set of problems we determine new estimates on the size of codes in the Johnson space. We also suggest a new approach to the derivation of the well-known Johnson bound for codes in this space. Our estimates are often valid in the region where the Johnson bound is vacuous. We show that these methods are also applicable to the case of multiple packings in the Hamming space (list-decodable codes). In this context we recover the best known estimate on the size of list-decodable codes in a new way

Canonical form for poset codes and coding-decoding schemes for expected loss

Orientador: Marcelo FirerTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: No contexto de códigos corretores de erros, métricas são utilizadas para definir decodificadores de máxima proximidade, uma alternativa aos decodificadores de máxima verossimilhança. A família de métricas poset tem sido extensivamente estudada no contexto de teoria de códigos. Considerando a estrutura do grupo de isometrias lineares, é obtida uma forma canônica para matrizes geradoras de códigos lineares. Esta forma canônica permite obter expressões e limitantes analíticos para alguns invariantes clássicos da teoria: raio de empacotamento e complexidade de síndrome. Ainda, substituindo a probabilidade de erro pela perda esperada definida pelo desvio médio quadrático (entre a informação original e a informação decodificada), definimos uma proposta de codificação com ordem lexicográfica que, em algumas situações é ótima e em outras, as simulações feitas sugerem um desempenho ao menos subótimo. Finalmente, relacionamos a medida de perda esperada com proteção desigual de erros, fornecendo uma construção de códigos com dois níveis de proteção desigual de erros e com perda esperada menor que a obtida pelo produto de dois códigos ótimos, que separam as informações que são protegidas de modo diferenciadoAbstract: In the context of error-correcting codes, metrics are used to define minimum distance decoders, an alternative to maximum likelihood decoders. The family of poset metrics has been extensively studied in the context of coding theory. Considering the structure of the group of linear isometries, we obtain a canonical form for generator matrices of linear codes. The canonical form allows to obtain analytics expressions and bounds for classical invariants of the theory: packing radius and syndrome complexity. By substituting the error probability by the expected loss defined by the mean square deviation (between the original information and the decoded information), we propose an encoder scheme which, in some situations is optimal, and in others the simulations suggest a performance at least sub-optimal. Finally, we relate the expected loss measure with unequal error protection, providing a construction of codes with two levels of unequal error protection and expected loss smaller than the one obtained by the product of two optimal codes, which divide the information that is protected differentlyDoutoradoMatematicaDoutor em Matemática141586/2014-1CNPQCAPE