69 research outputs found
Projective codes meeting the Griesmer bound
AbstractWe present a brief survey of projective codes meeting the Griesmer bound. Methods for constructing large families of codes as well as sporadic codes meeting the bound are given. Current research on the classification of codes meeting the Griesmer bound is also presented
Applications of finite geometry in coding theory and cryptography
We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how
finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory
A study of (x(q+1),x;2,q)-minihypers
In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x <= q(2) - q, not decomposable in the sum of another minihyper and a line, a (j (q + 1), j; 2, q)-minihyper, where j = q(2) - q-x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44: 169-196, 2007), giving further results on these minihypers
A study of (xvt,xvt−1)-minihypers in PG(t,q)
AbstractWe study (xvt,xvt−1)-minihypers in PG(t,q), i.e. minihypers with the same parameters as a weighted sum of x hyperplanes. We characterize these minihypers as a nonnegative rational sum of hyperplanes and we use this characterization to extend and improve the main results of several papers which have appeared on the special case t=2. We establish a new link with coding theory and we use this link to construct several new infinite classes of (xvt,xvt−1)-minihypers in PG(t,q) that cannot be written as an integer sum of hyperplanes
Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances
The purpose of this paper is two-fold. First, we characterize the existence
of binary self-orthogonal codes meeting the Griesmer bound by employing
Solomon-Stiffler codes and some related residual codes. Second, using such a
characterization, we determine the exact value of except for five
special cases and the exact value of except for 41 special cases,
where denotes the largest minimum distance among all binary
self-orthogonal codes. Currently, the exact value of was determined by Shi et al. (2022). In addition, we develop a general
method to prove the nonexistence of some binary self-orthogonal codes by
considering the residual code of a binary self-orthogonal code.Comment: Submitted 20 January, 202
An improvement of the Griesmer bound for some small minimum distances
AbstractIn this paper we give some lower and upper bounds for the smallest length n(k, d) of a binary linear code with dimension k and minimum distance d. The lower bounds improve the known ones for small d. In the last section we summarize what we know about n(8, d)
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