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Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an [ n , k , d ] Euclidean LCD code over F q with q > 3 exists if there is an [ n , k , d ] linear code over F q , where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters. [ABSTRACT FROM AUTHOR]

The classical way of extending an [ n , k , d ] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C ‾ (− 1) and called the standardly extended code of C , is a linear code with parameters [ n + 1 , k , d ¯ ] , where d ¯ = d or d ¯ = d + 1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we determine the Hamming weight distribution of several classes of double cyclic codes over the finite non-chain ring F q + u F q with u 2 = u , which is isomorphic to F q × F q. Some two-weight and three-weight linear codes over the finite field F q are constructed by double cyclic codes over this ring. [ABSTRACT FROM AUTHOR]

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S (2 , q + 1 , (q n − 1) / (q − 1)) is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the m -sequences with Welch-type decimation d = 2 ⋅ 3 (n − 1) / 2 + 1 , and obtain some infinite family of 2-designs and a family of Steiner systems S (2 , 4 , (3 n − 1) / 2) using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space PG (n − 1 , 3) and thus is a pseudo-geometric design. [ABSTRACT FROM AUTHOR]

Let p and q be odd primes and q be a cubic primitive modulo p u for some positive integer u. In this paper, we prove that the solutions of some Diophantine equations provide the weight enumerators of some cubic primitive irreducible cyclic codes of prime length. Bounds on the minimum distances of these codes are also given. [ABSTRACT FROM AUTHOR]

For a simple-root λ -constacyclic code C over F q , let 〈 ρ 〉 and 〈 ρ , M 〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ , and by the cyclic shift ρ and the scalar multiplication group M , respectively. Let N G (C ⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C ⁎ = C ﹨ { 0 }. In this paper, we establish explicit formulas for N 〈 ρ 〉 (C ⁎) and N 〈 ρ , M 〉 (C ⁎). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible λ -constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N 〈 ρ 〉 (C ⁎) = N 〈 ρ , M 〉 (C ⁎) is presented. [ABSTRACT FROM AUTHOR]