New extremal binary self-dual codes of length 64 from $$R_3$$ R 3 -lifts of the extended binary Hamming code

Abstract

In this paper, we use the graded ring construction to lift the extended binary Hamming code of length 8 to $$R_k$$ R k . Using this method we construct self-dual codes over $$R_3$$ R 3 of length 8 whose Gray images are self-dual binary codes of length 64. In this way, we obtain twenty six non-equivalent extremal binary Type I self-dual codes of length 64, ten of which have weight enumerators that were not previously known to exist. The new codes that we found have $$\beta = 1, 5, 13, 17, 21, 25, 29, 33, 41$$ β = 1 , 5 , 13 , 17 , 21 , 25 , 29 , 33 , 41 and 52 in $$W_{64,2}$$ W 64 , 2 and they all have automorphism groups of size 8.