2,417 research outputs found
Semidefinite bounds for mixed binary/ternary codes
For nonnegative integers and , let denote the
maximum cardinality of a code of length , with binary
coordinates and ternary coordinates (in this order) and with minimum
distance at least . For a nonnegative integer , let
denote the collection of codes of cardinality at most . For , define . Then is upper bounded by the maximum
value of , where is a function
such that and if has minimum distance less than , and such that the matrix is positive semidefinite for each
. By exploiting symmetry, the semidefinite programming
problem for the case is reduced using representation theory. It yields
new upper bounds that are provided in tablesComment: 12 pages; some typos have been fixed. Accepted for publication in
Discrete Mathematic
Computing Extensions of Linear Codes
This paper deals with the problem of increasing the minimum distance of a
linear code by adding one or more columns to the generator matrix. Several
methods to compute extensions of linear codes are presented. Many codes
improving the previously known lower bounds on the minimum distance have been
found.Comment: accepted for publication at ISIT 0
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 lengths of divisible codes
In this article, the effective lengths of all -divisible linear codes
over with a non-negative integer are determined. For that
purpose, the -adic expansion of an integer is introduced. It is
shown that there exists a -divisible -linear code of
effective length if and only if the leading coefficient of the
-adic expansion of is non-negative. Furthermore, the maximum weight
of a -divisible code of effective length is at most ,
where denotes the cross-sum of the -adic expansion of .
This result has applications in Galois geometries. A recent theorem of
N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a
corollary. Furthermore, we get an improvement of the Johnson bound for constant
dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An
improvement of the Johnson bound for subspace codes
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