Asymptotic performance of double circulant and four circulant codes with small hull dimension

Abstract We present enumeration formulas for double circulant (DC) codes of length $$2m$$ and four circulant (FC) codes of length $$4m$$ over the finite field $$\mathbb {F}_q$$ , with prescribed Euclidean hull dimension, assuming $$\gcd (m, q) = 1$$ . These formulas significantly generalize previous results that were limited to special cases. In particular, we resolve an open problem posed by Zhu and Shi (J. Appl. Math. Comput. 68:1227–1244, 2022) concerning the case of reciprocal pair factors in the factorization of $$x^m - 1$$ for the enumeration of self-dual and linear complementary dual (LCD) FC codes. Additionally, we correct an enumeration formula used in Zhu and Shi (Bull. Aust. Math. Soc. 98(1):159–166, 2018), leading to an improved bound on the relative distance for LCD FC codes.