New self-orthogonal codes from weakly regular plateaued functions and their application in LCD codes

A linear code with few weights is a significant code family in coding theory. A linear code is considered self-orthogonal if contained within its dual code. Self-orthogonal codes have applications in linear complementary dual codes, quantum codes, etc. The construction of linear codes is an interesting research problem. There are various methods to construct linear codes, and one approach involves utilizing cryptographic functions defined over finite fields. The construction of linear codes (in particular, self-orthogonal codes) from functions has been studied in the literature. In this paper, we generalize the construction method given by Heng et al. in [Des. Codes Cryptogr. 91(12), 2023] to weakly regular plateaued functions. We first construct several families of p-ary linear codes with few weights from weakly regular plateaued unbalanced (resp. balanced) functions over the finite fields of odd characteristics. We observe that the constructed codes are self-orthogonal codes when p = 3. Then, we use the constructed ternary self-orthogonal codes to build new families of ternary LCD codes. Consequently, we obtain (almost) optimal ternary self-orthogonal codes and LCD codes

Codes from orbit matrices of strongly regular graphs

We show that under certain conditions submatrices of orbit matrices of strongly regular graphs span self-orthogonal codes. In order to demonstrate this method of construction, we construct self-orthogonal binary linear codes from orbit matrices of the triangular graphs T(2k) with at most 120 vertices. Further, we obtain strongly regular graphs and block designs from codewords of the constructed codes

The hull of two classical propagation rules and their applications

Propagation rules are of great help in constructing good linear codes. Both Euclidean and Hermitian hulls of linear codes perform an important part in coding theory. In this paper, we consider these two aspects together and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for resulting codes derived from these two propagation rules being self-dual, self-orthogonal or linear complement dual (LCD) codes are given. As applications, we construct some linear codes with prescribed hull dimensions and many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD codes and good quaternary Hermitian LCD codes which are optimal or have best or almost best known parameters according to Datebase at $http://www.codetables.de$. Moreover, our methods contributes positively to improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table