Ternary primitive LCD BCH codes

TL;DR

This paper investigates ternary primitive BCH codes, identifying key coset leaders, constructing specific codes, and determining their weight distributions, including complex calculations related to Kloosterman sums.

Contribution

It extends the concept of absolute coset leaders to ternary codes and provides new classes of LCD BCH codes with known weight distributions.

Findings

Identified largest, second largest, and third largest absolute coset leaders for ternary primitive BCH codes.

Constructed three classes of ternary primitive BCH codes with known weight distributions.

Calculated weight distributions for certain LCD BCH codes, involving Kloosterman sums.

Abstract

Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums.