Characterization results on weighted minihypers and on linear codes meeting the G(r)iesmer bound

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The use of blocking sets in Galois geometries and in related research areas

Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems

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

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

The second Feng-Rao number for codes coming from telescopic semigroups

In this manuscript we show that the second Feng-Rao number of any telescopic numerical semigroup agrees with the multiplicity of the semigroup. To achieve this result we first study the behavior of Ap\'ery sets under gluings of numerical semigroups. These results provide a bound for the second Hamming weight of one-point Algebraic Geometry codes, which improves upon other estimates such as the Griesmer Order Bound

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 $d_{so}(n,7)$ except for five special cases and the exact value of $d_{so}(n,8)$ except for 41 special cases, where $d_{so}(n,k)$ denotes the largest minimum distance among all binary self-orthogonal $[n, k]$ codes. Currently, the exact value of $d_{so}(n,k)$ $(k \le 6)$ 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)