Type II codes over F/sub 2/+uF/sub 2/

Self-Dual Codes and Fixed-Point-Free Permutations of Order 2

“…We find binary optimal self-dual codes of length 52 in [3,4,10,12,21,29,31,30,32]. Also, codes over the ring F 2 + uF 2 have been also interesting due to their usefulness for construction of Hermitian modular forms [1] and Gaussian lattices [9]. We find all self-dual codes over F 2 + uF 2 of length up to 8 in [9].…”

“…We use a self-orthogonal [25,12,6] code with a generator matrix G 25 = (I 12 B) where I 12 is the identity matrix, and B is the 12 × 12 circulant matrix with the first row vector 1100101111101. We let r = 10 and M = {(1, 4), (2,20), (3,19), (5,21), (6,7), (8,9), (10,11), (17,18), (22,23), (24,25)}, then we obtain a self-dual [50, 25,10] code C 50,1 with weight enumerator W 2 for β = 0. The order of the automorphism group of C 50,1 is 2.…”

“…The order of the automorphism group of C 50,1 is 2. If we let r = 11 and M = {(1, 7), (2,12), (3,13), (6,8), (9,10), (14,15), (16,17), (18,19), (20,21), (22,23), (24,25)}, then we obtain an optimal self-dual code [50, 25,10] code C 50,2 with weight enumerator W 2 for β = 0. The order of the automorphism group of C 50,2 is 2.…”

“…Also, codes over the ring F 2 + uF 2 have been also interesting due to their usefulness for construction of Hermitian modular forms [1] and Gaussian lattices [9]. We find all self-dual codes over F 2 + uF 2 of length up to 8 in [9]. All Lee-extremal and Lee-optimal self-dual codes over F 2 + uF 2 of lengths 9 through 24 with a nontrivial automorphism of odd order are classified in [18,19,22,24,25].…”

“…If C is a self-dual code over the ring F 2 +uF 2 of length n, then the Gray image of C is a binary self-dual code of length 2n with a fixed-point-free automorphism of order 2 [9].…”