Linear code derived from the primage of quadratic function

Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power $q$, we construct some class of linear code over finite field $\mathbb{F}_q$ with defining set be the preimage of general quadratic form function and determine the explicit complete weight enumerators of the linear codes. Our construction cover all the corresponding result with quadratic form function and they may have applications in cryptography and secret sharing schemes.Comment: arXiv admin note: text overlap with arXiv:1506.06830 by other author

Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of $q$-ary linear codes under some certain conditions, where $q$ is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems

Linear Codes With Two or Three Weights From Some Functions With Low Walsh Spectrum in Odd Characteristic

Linear codes with few weights have applications in authentication codes, secrete sharing schemes, association schemes, consumer electronics and data storage system. In this paper, several classes of linear codes with two or three weights are obtained from some functions with low Walsh spectrum in odd characteristic. Numerical results show that some of the linear codes obtained are optimal or almost optimal in the sense that they meet certain bounds on linear codes.Comment: Some of the results of this paper are covered by others' wor