9 research outputs found
On the minimum number of minimal codewords
We study the minimum number of minimal codewords in linear codes from the
point of view of projective geometry. We derive bounds and in some cases
determine the exact values. We also present an extension to minimal subcode
supports.Comment: 8 pages, 1 tabl
A geometric characterization of minimal codes and their asymptotic performance
In this paper, we give a geometric characterization of minimal linear codes.
In particular, we relate minimal linear codes to cutting blocking sets,
introduced in a recent paper by Bonini and Borello. Using this
characterization, we derive some bounds on the length and the distance of
minimal codes, according to their dimension and the underlying field size.
Furthermore, we show that the family of minimal codes is asymptotically good.
Finally, we provide some geometrical constructions of minimal codes.Comment: 22 page
A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes
Boolean functions with high algebraic immunity are important cryptographic
primitives in some stream ciphers. In this paper, two methodologies for
constructing binary minimal codes from sets, Boolean functions and vectorial
Boolean functions with high algebraic immunity are proposed. More precisely, a
general construction of new minimal codes using minimal codes contained in
Reed-Muller codes and sets without nonzero low degree annihilators is
presented. The other construction allows us to yield minimal codes from certain
subcodes of Reed-Muller codes and vectorial Boolean functions with high
algebraic immunity. Via these general constructions, infinite families of
minimal binary linear codes of dimension and length less than or equal to
are obtained. In addition, a lower bound on the minimum distance of
the proposed minimal linear codes is established. Conjectures and open problems
are also presented. The results of this paper show that Boolean functions with
high algebraic immunity have nice applications in several fields such as
symmetric cryptography, coding theory and secret sharing schemes