ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.458601 J. Algebra Comb. Discrete Appl. 5(3) • 143–151 Received: 21 August 2017 Accepted: 13 June 2018 Journal of Algebra Combinatorics Discrete Structures and Applications New extremal singly even self-dual codes of lengths 64 and 66∗ Research Article Damyan Anev, Masaaki Harada, Nikolay Yankov Abstract: For lengths 64 and 66, we construct six and seven extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist, re- spectively. We also construct new 40 inequivalent extremal doubly even self-dual [64, 32, 12] codes with covering radius 12 meeting the Delsarte bound. These new codes are constructed by considering four-circulant codes along with their neighbors and shadows. 2010 MSC: 94B05 Keywords: Self-dual code, Weight enumerator 1. Introduction A (binary) [n,k] code C is a k-dimensional vector subspace of Fn2 , where F2 denotes the finite field of order 2. All codes in this note are binary. The parameter n is called the length of C. The weight wt(x) of a vector x is the number of non-zero components of x. A vector of C is a codeword of C. The minimum non-zero weight of all codewords in C is called the minimum weight of C. An [n,k] code with minimum weight d is called an [n,k,d] code. The dual code C⊥ of a code C of length n is defined as C⊥ = {x ∈ Fn2 | x · y = 0 for all y ∈ C}, where x · y is the standard inner product. A code C is called self-dual if C = C⊥. A self-dual code C is doubly even if all codewords of C have weight divisible by four, and singly even if there is at least one codeword x with wt(x) ≡ 2 (mod 4). It is known that a self-dual code of length n exists if and only if n is even, and a doubly even self-dual code of length n exists if and only if n is divisible by 8. Let C be a singly even self-dual code. Let C0 denote the subcode of C consisting of codewords x with wt(x) ≡ 0 (mod 4). The shadow S of C is defined to be C⊥0 \ C. Shadows for self-dual codes ∗ This work was supported by JSPS KAKENHI Grant Number 15H03633. Damyan Anev, Nikolay Yankov; Faculty of Mathematics and Informatics, Konstantin Preslavski University of Shumen, Shumen, 9712, Bulgaria (email: damian_anev@mail.bg, jankov_niki@yahoo.com). Masaaki Harada (Corresponding Author); Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan (email: mharada@tohoku.ac.jp). 143 https://orcid.org/0000-0002-3175-0168 https://orcid.org/0000-0002-2748-6456 https://orcid.org/0000-0003-3703-5867 D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 were introduced by Conway and Sloane [6] in order to give the largest possible minimum weight among singly even self-dual codes, and to provide restrictions on the weight enumerators of singly even self-dual codes. The largest possible minimum weights among singly even self-dual codes of length n were given for n ≤ 72 in [6]. The possible weight enumerators of singly even self-dual codes with the largest possible minimum weights were given in [6] and [7] for n ≤ 72. It is a fundamental problem to find which weight enumerators actually occur for the possible weight enumerators (see [6]). By considering the shadows, Rains [13] showed that the minimum weight d of a self-dual code of length n is bounded by d ≤ 4b n 24 c+6 if n ≡ 22 (mod 24), d ≤ 4b n 24 c+ 4 otherwise. A self-dual code meeting the bound is called extremal. The aim of this note is to construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. More precisely, we construct extremal singly even self-dual [64,32,12] codes with weight enumerators W64,1 for β = 35, and W64,2 for β ∈ {19,34,42,45,50} (see Section 2 for W64,1 and W64,2). These codes are constructed as self-dual neighbors of extremal four-circulant singly even self-dual codes. We construct extremal singly even self-dual [66,33,12] codes with weight enumerators W66,1 for β ∈ {7,58,70,91,93}, and W66,3 for β ∈{22,23} (see Section 2 for W66,1 and W66,3). These codes are constructed from extremal singly even self-dual [64,32,12] codes by the method given in [14]. We also demonstrate that there are at least 44 inequivalent extremal doubly even self-dual [64,32,12] codes with covering radius 12 meeting the Delsarte bound. All computer calculations in this note were done with the help of the algebra software Magma [1] and the computer system Q-extensions [2]. 2. Weight enumerators of extremal singly even self-dual codes of lengths 64 and 66 The possible weight enumerators W64,i and S64,i of extremal singly even self-dual [64,32,12] codes and their shadows are given in [6]:{ W64,1 = 1 + (1312 + 16β)y 12 + (22016−64β)y14 + · · · , S64,1 = y 4 + (β −14)y8 + (3419−12β)y12 + · · · ,{ W64,2 = 1 + (1312 + 16β)y 12 + (23040−64β)y14 + · · · , S64,2 = βy 8 + (3328−12β)y12 + · · · , where β are integers with 14 ≤ β ≤ 104 for W64,1 and 0 ≤ β ≤ 277 for W64,2. Extremal singly even self-dual codes with weight enumerator W64,1 are known for β ∈ { 14,16,18,20,22,24,25,26,28,29,30,32, 34,36,38,39,44,46,53,59,60,64,74 } (see [4], [10], [11] and [16]). Extremal singly even self-dual codes with weight enumerator W64,2 are known for β ∈ { 0,1, . . . ,41,44,48,51,52,56,58,64,65,72, 80,88,96,104,108,112,114,118,120,184 } \{19,31,34,39} (see [4], [10], [16] and [18]). The possible weight enumerators W66,i and S66,i of extremal singly even self-dual [66,33,12] codes 144 D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 and their shadows are given in [7]:{ W66,1 = 1 + (858 + 8β)y 12 + (18678−24β)y14 + · · · , S66,1 = βy 9 + (10032−12β)y13 + · · · ,{ W66,2 = 1 + 1690y 12 + 7990y14 + · · · , S66,2 = y + 9680y 13 + · · · ,{ W66,3 = 1 + (858 + 8β)y 12 + (18166−24β)y14 + · · · , S66,3 = y 5 + (β −14)y9 + (10123−12β)y13 + · · · , where β are integers with 0 ≤ β ≤ 778 for W66,1 and 14 ≤ β ≤ 756 for W66,3. Extremal singly even self-dual codes with weight enumerator W66,1 are known for β ∈{0,1, . . . ,92,94,100,101,115}\{4,7,58,70,91} (see [5], [8], [10], [17] and [18]). Extremal singly even self-dual codes with weight enumerator W66,2 are known (see [8] and [15]). Extremal singly even self-dual codes with weight enumerator W66,3 are known for β ∈{24,25, . . . ,92}\{65,68,69,72,89,91} (see [9], [10], [11] and [12]). 3. Extremal four-circulant singly even self-dual [64, 32, 12] codes An n×n circulant matrix has the following form:  r0 r1 r2 · · · rn−1 rn−1 r0 r1 · · · rn−2 ... ... ... ... r1 r2 r3 · · · r0   , so that each successive row is a cyclic shift of the previous one. Let A and B be n×n circulant matrices. Let C be a [4n,2n] code with generator matrix of the following form:( I2n A B BT AT ) , (1) where In denotes the identity matrix of order n and AT denotes the transpose of A. It is easy to see that C is self-dual if AAT + BBT = In. The codes with generator matrices of the form (1) are called four-circulant. Two codes are equivalent if one can be obtained from the other by a permutation of coordinates. In this section, we give a classification of extremal four-circulant singly even self-dual [64,32,12] codes. Our exhaustive search found all distinct extremal four-circulant singly even self-dual [64,32,12] codes, which must be checked further for equivalence to complete the classification. This was done by considering all pairs of 16 × 16 circulant matrices A and B satisfying the condition that AAT + BBT = I16, the sum of the weights of the first rows of A and B is congruent to 1 (mod 4) and the sum of the weights is greater than or equal to 13. Since a cyclic shift of the first rows gives an equivalent code, we may assume without loss of generality that the last entry of the first row of B is 1. Then our computer search shows that the above distinct extremal four-circulant singly even self-dual [64,32,12] codes are divided into 67 inequivalent codes. Proposition 3.1. Up to equivalence, there are 67 extremal four-circulant singly even self-dual [64,32,12] codes. We denote the 67 codes by C64,i (i = 1,2, . . . ,67). For the 67 codes C64,i, the first rows rA (resp. rB) of the circulant matrices A (resp. B) in generator matrices (1) are listed in Table 1. We verified that the codes C64,i have weight enumerator W64,2, where β are also listed in Table 1. 145 D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 Table 1. Extremal four-circulant singly even self-dual [64, 32, 12] codes Codes rA rB β C64,1 (0000001100111111) (0001011010101111) 0 C64,2 (0000010101111101) (0010011010111011) 0 C64,3 (0000011001101111) (0010110101011011) 0 C64,4 (0000000001011111) (0001001100101011) 8 C64,5 (0000000010101111) (0011011011110111) 8 C64,6 (0000000011010111) (0000100110011011) 8 C64,7 (0000000011010111) (0000101100010111) 8 C64,8 (0000000011010111) (0011101110101111) 8 C64,9 (0000000110111111) (0101101111111111) 8 C64,10 (0000001001011101) (0001000101011011) 8 C64,11 (0000001100011111) (0010101011011111) 8 C64,12 (0000001100011111) (0010111011011011) 8 C64,13 (0000001100111011) (0001101011101111) 8 C64,14 (0000001101111111) (0011101111011111) 8 C64,15 (0000010000111101) (0010111011011111) 8 C64,16 (0000010001011111) (0001110101101111) 8 C64,17 (0000010110111011) (0001101110001111) 8 C64,18 (0000000100011111) (0010111111110011) 16 C64,19 (0000000100111101) (0000101011000111) 16 C64,20 (0000000110010111) (0001001111111111) 16 C64,21 (0000000111001111) (0010101110111101) 16 C64,22 (0000000111001111) (0010110110111011) 16 C64,23 (0000001000101111) (0011101011110111) 16 C64,24 (0000001011100011) (0010101111110111) 16 C64,25 (0000001011100011) (0011011011111011) 16 C64,26 (0000010010011111) (0010110011101111) 16 C64,27 (0000011001101111) (0001001011011111) 16 C64,28 (0000011011011111) (0010010101011101) 16 C64,29 (0000011011100111) (0001011111001011) 16 C64,30 (0000011101111111) (0101101110110111) 16 C64,31 (0000101110111111) (0011101011110111) 16 C64,32 (0000000000100111) (0001011101101011) 24 C64,33 (0000000001011011) (0010010101101011) 24 C64,34 (0000000100111111) (0001001000101011) 24 C64,35 (0000000101001011) (0010010110011011) 24 C64,36 (0000000101001011) (0010011001011011) 24 C64,37 (0000000110111111) (0000001000100111) 24 C64,38 (0000001001111111) (0010101111001011) 24 C64,39 (0000001100011111) (0001010011111111) 24 C64,40 (0000001100011111) (0001110011110111) 24 C64,41 (0000010001011111) (0010101111001111) 24 C64,42 (0000010001101111) (0011001110101111) 24 C64,43 (0000010011101111) (0001011101100111) 24 C64,44 (0000010101010111) (0001010111101111) 24 C64,45 (0000010101010111) (0010110011111011) 24 C64,46 (0000010101110111) (0000101111110011) 24 C64,47 (0000010101110111) (0001011101101011) 24 C64,48 (0000011011110111) (0101101110111111) 24 C64,49 (0000000001001011) (0000111010110111) 32 C64,50 (0000000001100111) (0001001111100011) 32 146 D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 Table 1. Extremal four-circulant singly even self-dual [64, 32, 12] codes (continued) Codes rA rB β C64,51 (0000001010111011) (0001011111100111) 32 C64,52 (0000010101011111) (0001101111000111) 32 C64,53 (0000010101111101) (0010110010110111) 32 C64,54 (0000011010111111) (0000101110011101) 32 C64,55 (0000101011101011) (0001011111001011) 32 C64,56 (0000000000100111) (0001011010111011) 40 C64,57 (0000000010101101) (0001001011011011) 40 C64,58 (0000001000011101) (0000100101111011) 40 C64,59 (0000001110011111) (0001010111101101) 40 C64,60 (0000011000111111) (0001010111101101) 40 C64,61 (0000011011001111) (0000101010111111) 40 C64,62 (0000100111011111) (0001010101011011) 40 C64,63 (0000001001101011) (0001010011001101) 48 C64,64 (0000000001011011) (0001011000101111) 56 C64,65 (0000010111011111) (0010100101011011) 56 C64,66 (0000101110011101) (0001000101111111) 64 C64,67 (0000000001011111) (0001011111110111) 72 4. Extremal self-dual [64, 32, 12] neighbors of C64,i Two self-dual codes C and C′ of length n are said to be neighbors if dim(C∩C′) = n/2−1. Any self- dual code of length n can be reached from any other by taking successive neighbors (see [6]). Since every self-dual code C of length n contains the all-one vector 1, C has 2n/2−1 − 1 subcodes D of codimension 1 containing 1. Since dim(D⊥/D) = 2, there are two self-dual codes rather than C lying between D⊥ and D. If C is a singly even self-dual code of length divisible by 8, then C has two doubly even self- dual neighbors (see [3]). In this section, we construct extremal self-dual [64,32,12] codes by considering self-dual neighbors. For i = 1,2, . . . ,67, we found all distinct extremal singly even self-dual neighbors of C64,i, which are equivalent to none of the 67 codes. Then we verified that these codes are divided into 385 inequivalent codes D64,i (i = 1,2, . . . ,385). These codes D64,i are constructed as 〈(C64,j ∩〈x〉⊥),x〉. To save space, the values j, the supports supp(x) of x, the values (k,β) in the weight enumerators W64,k are listed in “http://www.math.is.tohoku.ac.jp/~mharada/Paper/64-SE-d12.txt” for the 385 codes. 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: Proposition 4.1. There is an extremal singly even self-dual [64,32,12] code with weight enumerator W64,1 for β = 35, and W64,2 for β ∈{19,34,42,45,50}. Now we consider the extremal doubly even self-dual neighbors of C64,i (i = 1,2,3). Since the shadow has minimum weight 12, the two doubly even self-dual neighbors C164,i and C 2 64,i are extremal doubly even self-dual [64,32,12] codes with covering radius 12 (see [4]). Thus, six extremal doubly even self- dual [64,32,12] codes with covering radius 12 are constructed. In addition, among the 385 codes D64,i (i = 1,2, . . . ,385), the 19 extremal singly even self-dual codes D64,j have shadow of minimum weight 12, where j ∈{1,2,12,19,22,33,44,58,66,68,84,95,108,115,136,143,191,240,254}. 147 http://www.math.is.tohoku.ac.jp/~mharada/Paper/64-SE-d12.txt D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 Table 2. Extremal singly even self-dual [64, 32, 12] neighbors Codes j supp(x) (k,β) D64,138 24 {1,2,3,38,42,43,45,46,48,54,56,57} (2,19) D64,270 49 {1,2,8,32,38,41,48,49,50,53,55,61} (1,35) D64,283 52 {1,2,4,33,36,37,41,43,46,51,61,64} (2,42) D64,293 56 {3,7,9,10,11,37,43,53,57,58,62,64} (2,34) D64,314 64 {6,8,26,37,38,40,43,46,48,59,61,63} (2,50) D64,329 65 {1,6,8,9,37,47,50,52,57,60,63,64} (2,45) D64,1 1 {4,7,9,34,38,40,45,46,47,50,51,53} (2,0) D64,2 1 {3,37,38,47,48,50,52,53,54,59,60,63} (2,0) D64,12 4 {2,4,5,16,17,38,40,46,56,57,60,62} (2,0) D64,19 4 {2,3,6,7,9,35,41,49,55,56,57,61} (2,0) D64,22 4 {2,33,34,35,38,39,42,45,48,52,61,62} (2,0) D64,33 6 {8,9,10,16,17,33,44,45,54,55,59,61} (2,0) D64,44 6 {1,3,6,33,36,38,39,45,47,55,57,59} (2,0) D64,58 8 {1,3,5,16,17,35,36,38,42,44,54,59} (2,0) D64,66 8 {4,6,9,34,36,39,41,42,48,51,57,63} (2,0) D64,68 8 {3,6,9,33,36,37,38,49,56,57,60,62} (2,0) D64,84 13 {1,4,5,35,37,38,41,44,53,60,61,62} (2,0) D64,95 13 {2,4,9,34,35,40,42,47,49,52,59,64} (2,0) D64,108 15 {2,16,17,37,43,48,49,52,54,57,58,64} (2,0) D64,115 16 {1,3,6,7,8,41,45,46,49,50,57,60} (2,0) D64,136 21 {3,16,17,33,34,37,42,44,47,51,52,56} (2,0) D64,143 26 {1,2,9,34,37,38,41,48,57,58,59,64} (2,0) D64,191 35 {1,2,6,8,10,33,37,46,54,59,60,63} (2,0) D64,240 47 {2,4,7,9,13,16,17,44,56,59,62,64} (2,0) D64,254 48 {1,2,5,7,8,35,36,37,45,47,49,63} (2,0) D64,14 4 {1,7,8,35,36,37,41,43,46,49,51,53} (1,14) D64,383 67 {1,33,34,36,37,38,40,41,47,49,50,53,55,59,61,63} (2,40) The constructions of the 19 codes D64,j are listed in Table 2. Their two doubly even self-dual neighbors D164,j and D 2 64,j are extremal doubly even self-dual [64,32,12] codes with covering radius 12. We verified that there are the following equivalent codes among the four codes in [4], the six codes C164,i, C 2 64,i and the 38 codes D164,j, D 2 64,j, where D264,22 ∼= D 2 64,68,D 2 64,33 ∼= D264,84,D 2 64,44 ∼= D264,95,D 2 64,136 ∼= D264,143, where C ∼= D means that C and D are equivalent, and there is no other pair of equivalent codes. Therefore, we have the following proposition. Proposition 4.2. There are at least 44 inequivalent extremal doubly even self-dual [64,32,12] codes with covering radius 12 meeting the Delsarte bound. In order to distinguish two doubly even neighbors D164,i and D 2 64,i (i = 68,84,95,143), we list in Table 3 the supports supp(x) for the 8 codes, where D164,i and D 2 64,i are constructed as 〈(D64,i∩〈x〉 ⊥),x〉. 148 D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 Table 3. Extremal doubly even self-dual [64, 32, 12] neighbors Codes supp(x) D164,68 {1,4,7,34,35,36,47,54,55,58,60,63} D264,68 {1,4,5,6,30,42,45,47,54,56,58,64} D164,84 {16,17,33,39,43,46,48,49,51,54,58,64} D264,84 {1,2,6,33,35,38,40,42,52,57,59,60} D164,95 {1,2,6,33,35,38,40,42,52,57,59,60} D264,95 {3,33,38,41,45,47,51,53,58,60,62,64} D164,143 {1,4,10,40,43,46,52,54,58,61,62,63} D264,143 {1,31,34,42,44,45,46,50,51,52,54,62} 5. Four-circulant singly even self-dual [64, 32, 10] codes and self- dual neighbors Using an approach similar to that given in Section 3, our exhaustive search found all distinct four- circulant singly even self-dual [64,32,10] codes. Then our computer search shows that the distinct four-circulant singly even self-dual [64,32,10] codes are divided into 224 inequivalent codes. Proposition 5.1. Up to equivalence, there are 224 four-circulant singly even self-dual [64,32,10] codes. We denote the 224 codes by E64,i (i = 1,2, . . . ,224). For the codes, the first rows rA (resp. rB) of the circulant matrices A (resp. B) in generator matrices (1) can be obtained from “http://www.math.is.tohoku.ac.jp/~mharada/Paper/64-4cir-d10.txt”. The following method for constructing self-dual neighbors was given in [4]. For C = E64,i (i = 1,2, . . . ,224), let M be a matrix whose rows are the codewords of weight 10 in C. Suppose that there is a vector x of even weight such that MxT = 1T . (2) Then C0 = 〈x〉⊥ ∩C is a subcode of index 2 in C. We have self-dual neighbors 〈C0,x〉 and 〈C0,x + y〉 of C for some vector y ∈ C \ C0, which have no codeword of weight 10 in C. When C has a self-dual neighbor C′ with minimum weight 12, there is a vector x satisfying (2) and we can obtain C′ in this way. For i = 1,2, . . . ,224, we verified that there is a unique vector satisfying (2) and C has two self-dual neighbors, where C0 is a doubly even [64,31,12] code. In this case, the two neighbors are automatically doubly even. Hence, we have the following: Proposition 5.2. There is no extremal singly even self-dual [64,32,12] neighbor of E64,i for i = 1,2, . . . ,224. 6. Extremal singly even self-dual [66, 33, 12] codes The following method for constructing singly even self-dual codes was given in [14]. Let C be a self-dual code of length n. Let x be a vector of odd weight. Let C0 denote the subcode of C consisting of all codewords which are orthogonal to x. Then there are cosets C1,C2,C3 of C0 such that C0 ⊥ = C0 ∪C1 ∪C2 ∪C3, where C = C0 ∪C2 and x + C = C1 ∪C3. It was shown in [14] that C(x) = (0,0,C0)∪ (1,1,C2)∪ (1,0,C1)∪ (0,1,C3) (3) 149 http://www.math.is.tohoku.ac.jp/~mharada/Paper/64-4cir-d10.txt D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 is a self-dual code of length n + 2. In this section, we construct new extremal singly even self-dual codes of length 66 using this construction from the extremal singly even self-dual [64,32,12] codes obtained in Sections 3 and 4. Our exhaustive search shows that there are 1166 inequivalent extremal singly even self-dual [66,33,12] codes constructed as the codes C(x) in (3) from the codes C64,i (i = 1,2, . . . ,67). 1157 codes of the 1166 codes have weight enumerator W66,1 for β ∈ {7,8, . . . ,92} \ {9,11}, 3 of them have weight enumerator W66,3 for β ∈ {30,49,54}, and 6 of them have weight enumerator W66,2. Extremal singly even self-dual [66,33,12] codes with weight enumerator W66,1 for β ∈ {7,58,70,91} are constructed for the first time. For the four weight enumerators W , as an example, codes C66,i with weight enumerators W are given (i = 1,2,3,4). We list in Table 4 the values β in W , the codes C and the vectors x = (x1,x2, . . . ,x32) of C(x) in (3), where xj = 1 (j = 33, . . . ,64). Table 4. Extremal singly even self-dual [66, 33, 12] codes Codes β W C (x1, . . . ,x32) C66,1 7 W66,1 C64,1 (01101101101010010111111010101100) C66,2 58 W66,1 C64,56 (00001101100000011000110000011100) C66,3 70 W66,1 C64,66 (00100110011011001001011100000010) C66,4 91 W66,1 C64,67 (00001110110111110000011101000010) D66,1 22 W66,3 D64,14 (10100011100100110111101010011111) D66,2 23 W66,3 D64,14 (10111100111100000100101000100011) D66,3 93 W66,1 D64,383 (10100101011110010011001101001101) By applying the construction given in (3) to D64,i, we found more extremal singly even self-dual [66,33,12] codes D66,j with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. For the codes D66,j, we list in Table 4 the values β in the weight enumerators W , the codes C and the vectors x = (x1,x2, . . . ,x32) of C(x) in (3), where xi = 1 (i = 33, . . . ,64). Hence, we have the following: Proposition 6.1. There is an extremal singly even self-dual [66,33,12] code with weight enumerator W66,1 for β ∈{7,58,70,91,93}, and weight enumerator W66,3 for β ∈{22,23}. Remark 6.2. The code D66,1 has the smallest value β among known extremal singly even self-dual [66,33,12] codes with weight enumerator W66,3. References [1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symb. Comput. 24(3–4) (1997) 235–265. [2] I. Bouyukliev, "About the code equivalence" in Advances in Coding Theory and Cryptology, NJ, Hackensack: World Scientific, 2007. [3] R. A. Brualdi, V. S. Pless, Weight enumerators of self–dual codes, IEEE Trans. Inform. Theory 37(4) (1991) 1222–1225. [4] N. Chigira, M. Harada, M. Kitazume, Extremal self-dual codes of length 64 through neighbors and covering radii, Des. Codes Cryptogr. 42(1) (2007) 93–101. [5] P. Çomak, J. L. Kim, F. Özbudak, New cubic self–dual codes of length 54,60 and 66, Appl. Algebra Engrg. Comm. Comput. 29(4) (2018) 303–312. [6] J. H. Conway, N. J. A. Sloane, A new upper bound on the minimal distance of self–dual codes, IEEE Trans. Inform. Theory 36(6) (1990) 1319–1333. 150 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1142/9789812772022_0009 https://doi.org/10.1142/9789812772022_0009 https://doi.org/10.1109/18.86979 https://doi.org/10.1109/18.86979 https://doi.org/10.1007/s10623-006-9018-5 https://doi.org/10.1007/s10623-006-9018-5 https://doi.org/10.1007/s00200-017-0343-x https://doi.org/10.1007/s00200-017-0343-x https://doi.org/10.1109/18.59931 https://doi.org/10.1109/18.59931 D. Anev et al. / J. Algebra Comb. Discrete Appl. 5(3) (2018) 143–151 [7] S. T. Dougherty, T. A. Gulliver, M. Harada, Extremal binary self–dual codes, IEEE Trans. Inform. Theory 43(6) (1997) 2036–2047. [8] M. Harada, T. Nishimura, R. Yorgova, New extremal self–dual codes of length 66, Math. Balkanica (N.S.) 21(1–2) (2007) 113–121. [9] S. Karadeniz, B. Yildiz, New extremal binary self–dual codes of length 66 as extensions of self–dual codes over Rk, J. Franklin Inst. 350(8) (2013) 1963–1973. [10] A. Kaya, New extremal binary self–dual codes of lengths 64 and 66 from R2–lifts, Finite Fields Appl. 46 (2017) 271–279. [11] A. Kaya, B. Yildiz, A. Pasa, New extremal binary self–dual codes from a modified four circulant construction, Discrete Math. 339(3) (2016) 1086–1094. [12] A. Kaya, B. Yildiz, I. Siap, New extremal binary self–dual codes from F4 + uF4–lifts of quadratic circulant codes over F4, Finite Fields Appl. 35 (2015) 318–329. [13] E. M. Rains, Shadow bounds for self–dual codes, IEEE Trans. Inform. Theory 44(1) (1998) 134–139. [14] H.-P. Tsai, Existence of certain extremal self–dual codes, IEEE Trans. Inform. Theory 38(2) (1992) 501–504. [15] H.-P. Tsai, Extremal self–dual codes of lengths 66 and 68, IEEE Trans. Inform. Theory 45(6) (1999) 2129–2133. [16] N. Yankov, Self–dual [62,31,12] and [64,32,12] codes with an automorphism of order 7, Adv. Math. Commun. 8(1) (2014) 73–81. [17] N. Yankov, M. H. Lee, M. Gürel, M. Ivanova, Self–dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory 61(3) (2015) 1188–1193. [18] N. Yankov, M. Ivanova, M. H. Lee, Self–dual codes with an automorphism of order 7 and s–extremal codes of length 68, Finite Fields Appl. 51 (2018) 17–30. 151 https://doi.org/10.1109/18.641574 https://doi.org/10.1109/18.641574 https://mathscinet.ams.org/mathscinet-getitem?mr=2350723 https://mathscinet.ams.org/mathscinet-getitem?mr=2350723 https://doi.org/10.1016/j.jfranklin.2013.05.015 https://doi.org/10.1016/j.jfranklin.2013.05.015 https://doi.org/10.1016/j.ffa.2017.04.003 https://doi.org/10.1016/j.ffa.2017.04.003 https://doi.org/10.1016/j.disc.2015.10.041 https://doi.org/10.1016/j.disc.2015.10.041 https://doi.org/10.1016/j.ffa.2015.05.004 https://doi.org/10.1016/j.ffa.2015.05.004 https://doi.org/10.1109/18.651000 https://doi.org/10.1109/18.119711 https://doi.org/10.1109/18.119711 https://doi.org/10.1109/18.782156 https://doi.org/10.1109/18.782156 http://dx.doi.org/10.3934/amc.2014.8.73 http://dx.doi.org/10.3934/amc.2014.8.73 https://doi.org/10.1109/TIT.2015.2396915 https://doi.org/10.1109/TIT.2015.2396915 https://doi.org/10.1016/j.ffa.2017.12.001 https://doi.org/10.1016/j.ffa.2017.12.001 Introduction Weight enumerators of extremal singly even self-dual codes of lengths 64 and 66 Extremal four-circulant singly even self-dual [64,32,12] codes Extremal self-dual [64,32,12] neighbors of C64,i Four-circulant singly even self-dual [64,32,10] codes and self-dual neighbors Extremal singly even self-dual [66, 33, 12] codes References