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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
Further results on covering codes with radius R and codimension tR + 1
The length function is the smallest possible length of a -ary linear code with codimension (redundancy) and covering
radius . Let be the smallest size of a -saturating set
in the projective space . There is a one-to-one
correspondence between codes and -saturating -sets in
that implies . In this work, for
, new asymptotic upper bounds on are obtained in the
following form:
The new bounds are essentially better than the known ones. For , a new
construction of -saturating sets in the projective space
, providing sets of small sizes, is proposed. The
codes, obtained by the construction, have minimum distance , i.e. they are almost MDS (AMDS) codes. These codes are taken as the
starting ones in the lift-constructions (so-called "-concatenating
constructions") for covering codes to obtain infinite families of codes with
growing codimension , .Comment: 24 pages. arXiv admin note: text overlap with arXiv:2108.1360
On regular sets of affine type in finite Desarguesian planes and related codes
In this paper, we consider point sets of finite Desarguesian planes whose
multisets of intersection numbers with lines is the same for all but one
exceptional parallel class of lines. We call such sets regular of affine type.
When the lines of the exceptional parallel class have the same intersection
numbers, then we call these sets regular of pointed type. Classical examples
are e.g. unitals; a detailed study and constructions of such sets with few
intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here
provide some general construction methods for regular sets and describe a few
infinite families. The members of one of these families have the size of a
unital and meet affine lines of in one of possible
intersection numbers, each of them congruent to modulo . As a
byproduct, we determine the intersection sizes of the Hermitian curve defined
over with suitable rational curves of degree and
we obtain -divisible codes with non-zero weights. We also
determine the weight enumerator of the codes arising from the general
constructions modulus some -powers.Comment: 16 pages/revised and improved versio
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