6 research outputs found
Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, monomials and
trinomials over finite fields with even characteristic are employed to
construct a number of families of binary cyclic codes. Lower bounds on the
minimum weight of some families of the cyclic codes are developed. The minimum
weights of other families of the codes constructed in this paper are
determined. The dimensions of the codes are flexible. Some of the codes
presented in this paper are optimal or almost optimal in the sense that they
meet some bounds on linear codes. Open problems regarding binary cyclic codes
from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687,
arXiv:1206.437
p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)
Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support
p-Adic estimates of Hamming weights in Abelian codes over Galois rings
A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Rotulamentos de codigos por grupos de simetrias
Orientadores: Sueli Irene Rodrigues Costa, Reginaldo Palazzo JrTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação CientificaResumo: A tese versa sobre questões relativas a grupos de simetrias de cĂłdigos e sua utilização no rotulamento destes cĂłdigos. Um cĂłdigo Ă© rotulável por um grupo G se este grupo age como grupo de simetrias de modo livre e transitivo; os rotulamentos sĂŁo as bijeções naturais entre o grupo e suas Ăłrbitas. A importância disto vem das isometrias associadas entre anĂ©is e cĂłdigos que vĂŞm sendo usadas para obtenção de novos exemplos a partir de construções já conhecidas. Neste trabalho utilizamos grupos de simetrias de cĂłdigos em dois problemas distintos: o primeiro, sobre extensões de cĂłdigos quaternários via isometrias entre anĂ©is e cĂłdigos em espaços de Hamming, e o segundo sobre cĂłdigos em grafos que incluem os espaços de Lee. Um dado interessante Ă© que todos os grupos envolvidos podem ser escritos como produto semi-direto de dois grupos simĂ©tricos ou de um grupo simĂ©trico por um grupo abeliano (mais especificamente, o produto Ă© o "wreath product" destes grupos). Na parte relativa a espaços de Hamming, os resultados principais sĂŁo a descrição dos cĂłdigos propelineares como Ăłrbitas de grupos de simetrias e suas relações com os cĂłdigos G-lineares; a demonstração da inexistĂŞncia de rotulamentos cĂclicos de espaços de Hamming em geral; a determinação dos grupos de simetrias dos cĂłdigos de Reed-Muller generalizados de primeira ordem e rotulamentos cĂclicos para estes cĂłdigos. A existĂŞncia destes rotulamentos Ă© conhecida de trabalhos anteriores, e aqui fornecemos uma descrição alternativa, a qual determina todos os rotulamentos no caso binário. AlĂ©m disso, mostramos que as simetrias que rotulam RM(l,m) nĂŁo se estendem a isometrias do espaço ambiente. Quanto aos cĂłdigos sobre grafos, os principais resultados sĂŁo a explicitação de relações entre cĂłdigos em grafos e ladrilhamentos do espaço euclidiano; a construção de um grupo rotulador nĂŁo-abeliano para uma famĂlia de espaços de Lee; e a descrição de todos os cĂłdigos perfeitos de Lee em dimensĂŁo 2, via a consideração do problema de ladrilhamentos associado (estendendo resultados clássicos sobre estes cĂłdigos)Abstract: This work deals with questions related to symmetry groups of codes and their use as code labelings. A code is labeled by a group G if this group acts freely and transitively as a group of symmetries; the labelings are the natural bijections between the group and its orbits. The importance of labelings comes from the associated isometries between rings and codes which have been used as a means of constructing new codes from old ones. In this work we use symmetry groups of codes in two different problems: the first one, on extensions of quaternary codes via isometries between rings and codes in Hamming spaces, and the second on codes in graphs that include Lee spaces. An interesting feature is that all the groups involved can be expressed as wreath products of two symmetric groups or of a symmmetric group and an abelian group. Concerning Hamming spaces, the main results are the description of propelinear codes as orbits of symmetry groups and the determination of its relationship with G-linear codes; the proof of the non-existence of cyclic labelings of general Hamming spaces; the determination of the symmetry groups of the generalized first-order Reed-Muller codes and of cyclic labelings for these codes. The existence of these labelings is known from previous works, but here we provide an alternative description that determines all the labelings in the binary case. In addition, we show that the symmetries that label RM (1, m) are not extendable to symmetries of the ambient space. With respect to codes on graphs, the main results are the establishment of the relations between codes on graphs and tesselations of euclidean space; the construction of a non-abelian labeling group for a family of Lee spaces; and the description of all linear perfect Lee codes in dimension two, via the associated tesselation (thus extending classical results on these codes)DoutoradoDoutor em Matemátic