Improving the BCH lower bound for the ternary negacyclic family in Theorem 5.4

Derive a significantly stronger lower bound on the minimum distance for the ternary negacyclic codes constructed in Theorem 5.4, which are defined by unions of consecutive odd 3-cyclotomic cosets and have length n = (3^p − 1)/2 for prime p; demonstrate that the BCH-based bound stated in Theorem 5.4 can be improved substantially.

Background

In Section 5, the authors construct an infinite family of ternary negacyclic codes at lengths n = (3^p − 1)/2 using defining sets formed by consecutive odd cyclotomic cosets. Example 5.1 shows that the actual BCH lower bound can exceed the general bound derived in Theorem 5.4 (e.g., d ≥ 22 versus a general bound yielding 12 when p = 5).

This empirical gap motivates the conjecture that the theoretical lower bound provided by Theorem 5.4 is not tight and can be significantly improved.