An important family of codes for data storage systems, cryptography, consumer electronics, and network coding for error control in digital communications are the so-called cyclic codes. This kind of linear codes are also important due to their efficient encoding and decoding algorithms. Because of this, cyclic codes have been studied for many years, however their complete weight distributions are known only for a few cases. The complete weight distribution has a wide range of applications in many research fields as the information it contains is of vital use in practical applications. Unfortunately, obtaining these distributions is in general a very hard problem that normally involves the evaluation of sophisticated exponential sums, which leaves this problem open for most of the cyclic codes. In this paper we determine, for any finite field IFq, the explicit factorization of any polynomial of the formxq+1-c, wherec∈ IFq*. Then we use this result to obtain, without the need to evaluate any kind of exponential sum, the complete weight distributions of a family of irreducible cyclic codes of dimension two over any finite field. As an application of our findings, we employ the complete weight distributions of some irreducible cyclic codes presented here to construct systematic authentication codes, showing that they are optimal or almost optimal.

Abstract A class of optimal three-weight $$[q^k-1,k+1,q^{k-1}(q-1)-1]$$ [ q k - 1 , k + 1 , q k - 1 ( q - 1 ) - 1 ] cyclic codes over $${\mathrm{I\!F}}_q$$ I F q , with $$k\ge 2$$ k ≥ 2 , achieving the Griesmer bound, was presented by Heng and Yue (IEEE Trans Inf Theory 62(8):4501–4513, 2016. https://doi.org/10.1109/TIT.2016.2550029). In this paper we study some of the subfield codes of this class of optimal cyclic codes when $$k=2$$ k = 2 . The weight distributions of the subfield codes are settled. It turns out that some of these codes are optimal and others have the best known parameters. The duals of the subfield codes are also investigated and found to be almost optimal with respect to the sphere-packing bound. In addition, the covering structure for the studied subfield codes is determined. Some of these codes are found to have the important property that any nonzero codeword is minimal, which is a desirable property that is useful in the design of a secret sharing scheme based on a linear code. Moreover, a specific example of a secret sharing scheme based on one of these subfield codes is given. Finally, a class of optimal two-weight linear codes over $${\mathrm{I\!F}}_q$$ I F q , achieving the Griesmer bound, whose duals are almost optimal with respect to the sphere-packing bound is presented. Through a different approach, this class of optimal two-weight linear codes was reported very recently by Heng (IEEE Trans Inf Theory 69(2):978–994, 2023. https://doi.org/10.1109/TIT.2022.3203380). Furthermore, it is shown that these optimal codes can be used to construct strongly regular graphs.

Abstract Up to a new invariant $$\mu (b)$$ μ ( b ) , the complete b-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. (Des Codes Cryptogr 90(5):1113–1125, 2022). The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes.

A remarkably general result which provides the evaluation of a family of exponential sums was presented by Moisio in 2000. In this work, we use such a general result in order to determine the value distribution of a particular kind of exponential sum. Then, motivated by some new ideas of Ma, , we use this value distribution in order to formulate a general description for the weight distribution of a particular kind of reducible cyclic codes that has been recently studied. As will be shown, such a general description not only gives a unified explanation for these families of codes, but also gives the weight distribution for other reducible cyclic codes that have not been studied so far. In addition, among the kind of codes studied here, we will characterize those that are projective.

A class of two-weight non-irreducible cyclic codes was recently presented by Ma . This class of codes generalizes and extends some of the codes presented in papers by Wolfmann in 2005 and Vega and Wolfman in 2007. The aim of this note is to show that the class of codes presented by Ma is properly contained in a superclass of two-weight cyclic codes that was introduced by Vega in 2008.

The weight distribution of a class of nonirreducible cyclic codes was given by Ma. By imposing a special set of conditions on this class of codes, it was recently shown that the resulting codes, for this class, can be obtained as elements in a family of codes introduced by Vega in. Now, by using a different set of conditions, the aim of this paper is to present an extended version for the class of codes studied by Ma

For any integer $k \geq 1$, an isometry between codes over $\mathbb{Z}_{2^{k+1}}$ and codes over $\mathbb{Z}_4$ is defined and used to give an equivalent generalization of the Gray map to the one introduced in [C. Carlet, IEEE Trans. Inform. Theory, 44 (1998), pp. 1543-1547]. Several results related to the linearity or nonlinearity of codes over $\mathbb{Z}_{2^{k+1}}$, as well as its corresponding images under this map, are given. These results are similar to those presented in Theorems 4, 5, and 6 of [A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, IEEE Trans. Inform. Theory, 40 (1994), pp. 301-319] for codes over $\mathbb{Z}_4$.