2012/01/27 by Lee, Yoonjin, Kim, Jon-Lark
#05E99 #11T71 (secondary) #94B05 (Primary) #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1201.5689
We complete the building-up construction for self-dual codes by resolving the open cases over GF(q) with q ≡ 3 \pmod 4, and over \Zpm and Galois rings \GR(pm,r) with an odd prime p satisfying p ≡ 3 \pmod 4 with r odd. We also extend the building-up construction for self-dual codes to finite chain rings. Our building-up construction produces many new interesting self-dual codes. In particular, we construct 945 new extremal self-dual ternary [32,16,9] codes, each of which has a trivial automorphism group. We also obtain many new self-dual codes over \mathbb Z9 of lengths 12, 16, 20 all with minimum Hamming weight 6, which is the best possible minimum Hamming weight that free self-dual codes over \Z9 of these lengths can attain. From the constructed codes over \mathbb Z9, we reconstruct optimal Type I lattices of dimensions 12, 16, 20, and 24 using Construction A; this shows that our building-up construction can make a good contribution for finding optimal Type I lattices as well as self-dual codes. We also find new optimal self-dual [16,8,7] codes over GF(7) and new self-dual codes over GF(7) with the best known parameters [24,12,9].