5 research outputs found
Several families of ternary negacyclic codes and their duals
Constacyclic codes contain cyclic codes as a subclass and have nice algebraic
structures. Constacyclic codes have theoretical importance, as they are
connected to a number of areas of mathematics and outperform cyclic codes in
several aspects. Negacyclic codes are a subclass of constacyclic codes and are
distance-optimal in many cases. However, compared with the extensive study of
cyclic codes, negacyclic codes are much less studied. In this paper, several
families of ternary negacyclic codes and their duals are constructed and
analysed. These families of negacyclic codes and their duals contain
distance-optimal codes and have very good parameters in general
Twisted skew -codes
In this paper we investigate left ideals as codes in twisted skew group
rings. The considered rings, which are often algebras over a finite field,
allows us to detect many of the well-known codes. The presentation, given here,
unifies the concept of group codes, twisted group codes and skew group codes
Complementary Dual Codes for Counter-measures to Side-Channel Attacks
We recall why linear codes with complementary duals (LCD codes) play a role in counter-measures to passive and active side-channel analyses on embedded cryptosystems. The rate and the minimum distance of such LCD codes must be as large as possible. We investigate primary constructions of such codes, in particular with cyclic codes, specifically with generalized residue codes, and we study their idempotents. We study those secondary constructions which preserve the LCD property, and we characterize conditions under which codes obtained by puncturing, shortening or extending codes, or obtained by the Plotkin sum, can be LCD
Noncommutative coding theory and algebraic sets for skew PBW extensions
The classical commutative coding theory has been recently extended to
noncommutative rings of polynomial type. There are many interesting works in
coding theory over single Ore extensions. In this review article we present the
most relevant algebraic tools and properties of single Ore extensions used in
noncommutative coding theory. The last section represents the novelty of the
paper. We will discuss the algebraic sets arising in noncommutative coding
theory but for skew extensions. These extensions conform a general class
of noncommutative rings of polynomial type and cover several algebras arising
in physics and noncommutative algebraic geometry, in particular, they cover the
Ore extensions of endomorphism injective type and the polynomials rings over
fields.Comment: arXiv admin note: text overlap with arXiv:2106.1208