Linear code derived from the primage of quadratic function

TL;DR

This paper constructs a new class of linear codes over finite fields using quadratic form preimages and determines their weight enumerators, with potential applications in cryptography and secret sharing.

Contribution

It introduces a novel construction of linear codes based on quadratic form preimages and explicitly determines their weight enumerators.

Findings

Complete weight enumerators of the codes are explicitly determined.

The construction generalizes previous results related to quadratic form functions.

Potential applications in cryptography and secret sharing are identified.

Abstract

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.