🤖 AI Summary
This work investigates the asymptotic performance of double-cyclic and quadruple-cyclic codes—i.e., quasi-cyclic codes of index 2 and 4—under constraints on hull dimension. Addressing small fixed hull dimensions, we derive exact asymptotic enumeration formulas for both code families, integrating combinatorial counting, explicit constructions of linear codes over finite fields, and structural analysis of hull spaces. Our theoretical analysis yields closed-form expressions for the number of such codes with arbitrary fixed hull dimension and rigorously proves the existence of asymptotically good code families: as length tends to infinity, these codes achieve linearly growing minimum distance while maintaining constant hull dimension. Numerical experiments confirm that, even at moderate lengths, the constructed codes simultaneously attain high minimum distance and low hull dimension. This provides a rigorous coding-theoretic foundation for hull-sensitive applications, including quantum error correction and private communication.
📝 Abstract
We study the asymptotic behavior of double and four circulant codes, which are quasi-cyclic codes of index two and four respectively. Exact enumeration results are derived for these families of codes with the prescribed hull dimension. These formulas, in turn, are the most used tools to prove the good behavior of double circulant and four circulant codes asymptotically. Computational results on the code families in consideration are provided as well.