New results on odd prime field codes with self-orthogonality and weight analysis
Further Study of p-Ary Self-Orthogonal Codes From Square Functions
Information Theory
Summary
This paper studies special error-correcting codes created from certain mathematical functions over fields with an odd number of elements. The authors analyze these codes by examining their defining quadratic conditions and counting related sums, which lets them find the code parameters and how weights are distributed. They also define smaller codes by removing some parts and check their properties, including when these codes are self-orthogonal (a useful symmetry). Their work applies broadly to codes over any odd prime field and for all extension sizes.
What this means in practice
- •For communications engineers: Design error-correcting codes with guaranteed self-orthogonality and known weight properties for data transmission over noisy channels.
- •For cryptography developers: Build cryptographic primitives that require codes with specific dual-distance and divisibility features for enhanced security.
Authors
Tingting Wu, Xianhong Xie
Abstract
In this paper, we investigate a family of trace codes over finite fields of odd characteristic defined by two quadratic trace conditions. By analyzing the quadratic forms associated with these trace conditions and evaluating the corresponding character sums, we obtain the parameters and complete weight distributions of the codes. We further exploit the natural multiplicative orbit structure of the defining sets to obtain punctured codes and determine their parameters, weight distributions and dual distances. Moreover, we establish self?orthogonality criteria and investigate divisibility and projectivity of the constructed codes. These results give a unified treatment for arbitrary odd prime fields and for extension degrees of both parities.