Improving lower bounds for negacyclic families in Theorems 9.1 and 9.2

Develop significantly stronger lower bounds on the minimum distance for the q-ary negacyclic code families constructed in Theorems 9.1 (for q ≡ 3 mod 4) and 9.2 (for q ≡ 1 mod 4), beyond the current BCH-based bounds stated in those theorems; ascertain improved bounds by leveraging the structure of consecutive odd q-cyclotomic cosets used in their defining sets.

Background

Section 9 generalizes the cyclotomic-coset-based construction to q-ary negacyclic codes for two cases: q ≡ 3 mod 4 and q ≡ 1 mod 4, both at lengths n = (q^p − 1)/(q − 1) for prime p. The authors derive BCH-type lower bounds for these families.

Example 9.1 exhibits that actual bounds can be higher than the stated general bound (e.g., d ≥ 17 versus a general bound of 10), prompting the conjecture that the general lower bounds in Theorems 9.1 and 9.2 are improvable.