New extremal binary self-dual codes from a modified four circulant construction

In this work, we propose a modified four circulant construction for self-dual codes and a bordered version of the construction using the properties of λ-circulant and λ-reverse circulant matrices. By using the constructions on F₂, we obtain new binary codes of lengths 64 and 68. We also apply the constructions to the ring R₂ and considering the F₂ and R₁-extensions, we obtain new singly-even extremal binary self-dual codes of lengths 66 and 68. More precisely, we find 3 new codes of length 64, 13 new codes of length 66 and 21 new codes of length 68. These codes all have weight enumerators with parameters that were not known to exist in the literature.