Some constructions of \(l\)-Galois LCD codes (Q6646797)

Let \({\mathbb F}_q\) be the finite field with \(q = p^m\) elements. The Galois inner product is a generalization of Hermitian and Euclidean inner products. It's defined as \(a\ast_ l b= \sum\limits_{i=1}^{n}a_ib_i^{p^l},\) where \(0\leq l\leq m-1.\) Similarly, we can define the \(l\)-Galois dual code of \(C\) as \(C^{\bot_l}=\{a\in \mathbb{F}_q^n\vert c\ast_l a=0, \forall c\in C\}.\) This paper is dedicated to \(l\)-Galois codes, defined as codes satisfying \(C\cap C^{\bot_l}=\{0\}.\) Some constructions of Galois LCD codes over \(\mathbb{F}_q\) are provided. Certain examples are given, and the codes are defined using their generator matrices.\N\NFor these constructions, a matrix \(A\) for which \(A[\sigma^{ m-l}(A)]^t = I,\) \(0\leq l\leq m - 1\) where \(\sigma\) is the Frobenius map, is used. Furthermore, the Galois self-dual basis is defined and used in order to show the existence of a Euclidean LCD code from a linear code over some finite field. A useful application of these Galois LCD codes in a multi-secret sharing scheme is shown. Some codes with good error-correcting capability (that are optimal and almost optimal Galois LCD codes) are constructed as an illustration of the importance of the results presented. The newly constructed examples for \(l\)-Galois LCD codes are shown in a table. Some of the newly found codes are: for \(p=3, l=0,\) there are optimal, almost optimal, and almost MDS codes; for \(p=2, l=0\) or 1, there are MDS codes; for \(l=1\), \(p=3\) or 5, the codes constructed are almost MDS; for \(p=l=3,\) the codes constructed are MDS.