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Recently, subfield codes of geometric codes over large finite fields ${\mathrm {GF}}(q)$ with dimension 3 and 4 were studied and distance-optimal subfield codes over ${\mathrm {GF}}(p)$ were obtained, where $q=p^{m}$. The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of $[q+1, 2, q]$ MDS codes over ${\mathrm {GF}}(q)$ , and then studies the subfield codes over ${\mathrm {GF}}(p)$ of some of the $[q+1, 2,q]$ MDS codes over ${\mathrm {GF}}(q)$. Two families of dimension-optimal codes over ${\mathrm {GF}}(p)$ are obtained, and several families of nearly optimal codes over ${\mathrm {GF}}(p)$ are produced. Several open problems are also proposed in this paper. [ABSTRACT FROM AUTHOR]

If a linear code is contained in its dual, then it is said to be self-orthogonal. Self-orthogonal codes are an important subclass of linear codes which have many nice applications. In this paper, we mainly study the augmented codes of the linear codes constructed by Hu et al. (2022) [21] , and the subcodes of the codes punctured from the irreducible cyclic codes constructed by Ding and Yang (2013) [11]. The parameters and weight distributions of these codes are explicitly determined. These codes are proved to be self-orthogonal under certain conditions. As an application, these self-orthogonal codes are used to derive optimal or almost optimal quantum codes with new parameters. [ABSTRACT FROM AUTHOR]

Ovoids in ${\mathrm {PG}}(3, {\mathrm {GF}}(q))$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families of ovoids are known. The first is the elliptic quadrics and the second is the Tits ovoids. It is known that an ovoid in ${\mathrm {PG}}(3, {\mathrm {GF}}(q))$ corresponds to a $[q^{2}+1, 4, q^{2}-q]$ code over ${\mathrm {GF}}(q)$ , which is called an ovoid code. The objectives of this paper are to develop the general theories of subfield codes and to study the subfield codes of the two families of ovoid codes. The dimensions, minimum weights, and the weight distributions of the subfield codes of the elliptic quadric codes and Tits ovoid codes are settled. The parameters of the duals of these subfield codes are also studied. Some of the codes presented in this paper are optimal, and some are distance-optimal. The parameters of the subfield codes are new. [ABSTRACT FROM AUTHOR]

In this paper, we mainly determine the weight distributions of three classes of linear codes. Firstly, we prove that two classes of ternary linear codes from the following two planar functions have two or three weights: f ( x ) = x 3 k + 1 2 , x ∈ F 3 m , where k , m are odd, gcd ( m , k ) = 1 , and f ( x ) = x 3 k + 1 , x ∈ F 3 m , where m gcd ( m , k ) is odd. They are exactly a part of the open problem in Ding and Ding (2015 Section IV). Secondly, we construct a new class of binary linear codes with three weights. In particular, the linear codes in this paper have applications in consumer electronics, communication and secret sharing schemes. [ABSTRACT FROM AUTHOR]

Cyclic codes with a few weights are very useful in the design of frequency hopping sequences and the development of secret sharing schemes. In this paper, we mainly use Gauss sums to represent the Hamming weights of cyclic codes whose duals have two zeroes. A lower bound of the minimum Hamming distance is determined. In some cases, we give the Hamming weight distributions of the cyclic codes. In particular, we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound, which generalizes a Vega’s result, and several classes of cyclic codes with a few weights, which solve an open problem proposed by Vega. [ABSTRACT FROM AUTHOR]

In this letter, we use Gauss sums of index 2 to construct a class of new binary linear codes with at most three weights. In special cases, they are optimal or near optimal codes. These codes can be used in authentication codes and secret sharing schemes. [ABSTRACT FROM PUBLISHER]