1,575 research outputs found
On the Error-Correcting Performance of some Binary and Ternary Linear Codes
In this work, we determine the coset weight spectra of all binary cyclic codes of lengths up to 33, ternary cyclic and negacyclic codes of lengths up to 20 and of some binary linear codes of lengths up to 33 which are distance-optimal, by using some of the algebraic properties of the codes and a computer assisted search. Having these weight spectra the monotony of the function of the undetected error probability after t-error correction P(t)ue (C,p) could be checked with any precision for a linear time. We have used a programm written in Maple to check the monotony of P(t)ue (C,p) for the investigated codes for a finite set of points of p € [0, p/(q-1)] and in this way to determine which of them are not proper
Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes
Constructions of distance-optimal codes and quasi-perfect codes are
challenging problems and have attracted many attentions. In this paper, we give
the following three results.
1) If and , an infinite family of
distance-optimal -ary cyclic sum-rank codes with the block length
, the matrix size , the cardinality
and the minimum sum-rank distance four is constructed.
2) Block length and the matrix size distance-optimal
sum-rank codes with the minimum sum-rank distance four and the Singleton defect
four are constructed. These sum-rank codes are close to the sphere packing
bound , the Singleton-like bound and have much larger block length
.
3) For given positive integers satisfying , an infinite family
of quasi-perfect sum-rank codes with the matrix size , and the
minimum sum-rank distance three is also constructed. Quasi-perfect binary
sum-rank codes with the minimum sum-rank distance four are also given.
Almost MSRD -ary codes with the block lengths up to are given. We
show that more distance-optimal binary sum-rank codes can be obtained from the
Plotkin sum.Comment: 19 pages, only quasi-perfect sum-rank codes were constructed. Almost
MSRD codes with the block lengths up to were include
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
On the Weight Distribution of the Coset Leaders of Constacyclic Codes
Constacyclic codes with one and the same generator polynomial and distinct length are considered. We give a generalization of the previous result of the first author [4] for constacyclic codes. Suitable maps between
vector spaces determined by the lengths of the codes are applied. It is proven
that the weight distributions of the coset leaders don’t depend on the word
length, but on generator polynomials only. In particular, we prove that every
constacyclic code has the same weight distribution of the coset leaders as a
suitable cyclic code
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
How many weights can a cyclic code have ?
Upper and lower bounds on the largest number of weights in a cyclic code of
given length, dimension and alphabet are given. An application to irreducible
cyclic codes is considered. Sharper upper bounds are given for the special
cyclic codes (called here strongly cyclic), {whose nonzero codewords have
period equal to the length of the code}. Asymptotics are derived on the
function {that is defined as} the largest number of nonzero
weights a cyclic code of dimension over \F_q can have, and an algorithm
to compute it is sketched. The nonzero weights in some infinite families of
Reed-Muller codes, either binary or -ary, as well as in the -ary Hamming
code are determined, two difficult results of independent interest.Comment: submitted on 21 June, 201
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