Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7

Some Restrictions on Weight Enumerators of Singly Even Self-Dual Codes II

“…respectively, where β is an integer with 0 ≤ β ≤ 93. Table 1 gives the following: It is known that there exists an extremal singly even self-dual [62, 31, 12] code with weight enumerator W 62 for β = 0, 2, 9, 10, 15, 16 (see [13]).…”

New extremal binary self-dual codes from F4+uF4-lifts of quadratic circulant codes over Kaya1, Yıldız2, Şiap3 2015Finite Fields and Their Applications

“…where A i is the circulant matrix with first row [20] and all the codes in Table 2 have an automorphism group of order 2 2 7.…”

New extremal singly even self-dual codes of lengths 64 and 66

“…The possible weight enumerators W 64,i and S 64,i of extremal singly even selfdual [64,32,12] codes and their shadows are given in [6]: (see [4], [10], [11] and [16] (see [4], [10], [16] and [18]). The possible weight enumerators W 66,i and S 66,i of extremal singly even self-dual [66, 33, 12] codes and their shadows are given in [7]: W 66,1 = 1 + (858 + 8β)y 12 + (18678 − 24β)y 14 + · · · , S 66,1 = βy 9 + (10032 − 12β)y 13 + · · · , W 66,2 = 1 + 1690y 12 + 7990y 14 + · · · , S 66,2 = y + 9680y 13 + · · · , W 66,3 = 1 + (858 + 8β)y 12 + (18166 − 24β)y 14 + · · · , S 66,3 = y 5 + (β − 14)y 9 + (10123 − 12β)y 13 + · · · , where β are integers with 0 ≤ β ≤ 778 for W 66,1 and 14 ≤ β ≤ 756 for W 66,3 .…”

“…For extremal singly even self-dual [64, 32, 12] codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist, j, supp(x) and (k, β) are list in Table 2. Hence, we have the following: (0000010101111101) (0010011010111011) 0 C 64, 3 (0000011001101111) (0010110101011011) 0 C 64, 4 (0000000001011111) (0001001100101011) 8 C 64, 5 (0000000010101111) (0011011011110111) 8 C 64, 6 (0000000011010111) (0000100110011011) 8 C 64, 7 (0000000011010111) (0000101100010111) 8 C 64, 8 (0000000011010111) (0011101110101111) 8 C 64, 9 (0000000110111111) (0101101111111111) 8 C 64, 10 (0000001001011101) (0001000101011011) 8 C 64, 11 (0000001100011111) (0010101011011111) 8 C 64, 12 (0000001100011111) (0010111011011011) 8 C 64, 13 (0000001100111011) (0001101011101111) 8 C 64, 14 (0000001101111111) (0011101111011111) 8 C 64, 15 (0000010000111101) (0010111011011111) 8 C 64, 16 (0000010001011111) (0001110101101111) 8 C 64, 17 (0000010110111011) (0001101110001111) 8 C 64, 18 (0000000100011111) Now we consider the extremal doubly even self-dual neighbors of C 64,i (i = 1, 2, 3). Since the shadow has minimum weight 12, the two doubly even self-dual neighbors C 1 64,i and C 2 64,i are extremal doubly even self-dual [64, 32, 12] codes with covering radius 12 (see [4]).…”