Weight congruences for p-ary cyclic codes

Several classes of optimal p-ary cyclic codes with minimum distance four

A lemma on polynomials modulo [formula omitted] and applications to coding theory

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general, the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li studied a class of p-ary cyclic codes of length p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> -1, where p is a prime and m is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of p-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even m and odd m along with the idea of Li The weight distributions of two related families of cyclic codes are also determined.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of p-ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.

As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented.

A class of power functions with four-valued Walsh transform and related cyclic codes

The weight distributions of two classes of p-ary cyclic codes with few weights

The weight distributions of two classes of p-ary cyclic codes

New classes of optimal p-ary cyclic codes with minimum distance four

The weight distribution of a class of p-ary cyclic codes

For any odd prime pź5, some optimal p-ary cyclic codes with parameters [pmź1,pmź2mź2,4] are presented by using perfect nonlinear monomials and the inverse function over źźpm$\mathbb {F}_{p^{m}}$. In addition, almost perfect nonlinear monomials, and other monomials over źź5m$\mathbb {F}_{5^{m}}$ are used to construct optimal quinary cyclic codes with parameters [5mź1,5mź2mź2,4].

Let n, k, e, m be positive integers such that n ≥ 3, 1 ≤ k ≤ n - 1, gcd(n, k) = e, and $m={n\\over e}$ is odd. In this paper, for an odd prime p, we derive a lower bound for the minimal distance of a large class of p-ary cyclic codes Cl with nonzeros α-1, α-(pk+1), α-(p3k+1), …, α-(p(2l-1)k+1), where $1\\leq l\\leq {m-1\\over 2}$ and α is a primitive element of the finite field Fpn. Employing these codes, p-ary sequence families with a flexible tradeoff between low correlation and large size are constructed.

In this paper, we investigate a class of p-ary cyclic codes whose duals have two zeros for some special cases and calculate their weight distributions explicitly. The results show that the codes have at most five nonzero weights. Moreover, they contain some optimal three-weight codes meeting the Griesmer bound.