New binary self-dual codes of lengths 50 to 60

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1 Designs, Codes and Cryptography manuscript No. (will be inserted by the editor) New binary self-dual codes of lengths 50 to 60 Nikolay Yankov Moon Ho Lee Received: date / Accepted: date Abstract Using a method for constructing binary self-dual codes with an automorphism of odd prime order p, we give a full classification of all optimal binary self-dual [50 + 2t, 25 + t] codes having an automorphism of order 5 for t = 0,..., 5. As a consequence, we determine the weight enumerators for which there is an optimal binary self-dual [52, 26, 10] code. Some of the constructed codes for lengths 52, 54, 58, and 60 have new values for the parameter in their weight enumerator. We also construct more than 3000 new doubly-even [56, 28, 12] self-dual codes. Keywords Self-dual codes Automorphism Classification Mathematics Subject Classification (2000) 94B05 11T71 1 Introduction Let F q be the finite field of q elements, for a prime power q. A linear [n, k] q code C is a k-dimensional subspace of F n q. The elements of C are called codewords, and the (Hamming) weight of a codeword v C is the number of the nonzero coordinates of v. We use wt(v) to denote the weight of a codeword. The minimum weight d of C is the minimum nonzero weight of any codeword in C and the code is called an [n, k, d] q code. A matrix whose rows form a basis of C is called a generator matrix of this code. Let (u, v) F q for u, v F n q be an inner product in F n q. The dual code of an [n, k] q code C is C = {u F n q (u, v) = 0 for all v C} and C is a N. Yankov Faculty of Mathematics and Informatics, Shumen University, 9700 Shumen, Bulgaria jankov niki@yahoo.com M. Lee Corresponding Author Division of Electronics & Information Engineering, Chonbuk National University, GA Dekjin-Dong, Jeonju City, Jeonbuk, South Korea

2 2 Nikolay Yankov, Moon Ho Lee linear [n, n k] q code. In the binary case the inner product is the standard one, namely, (u, v) = n i=1 u iv i. If C C, C is termed self-orthogonal, and if C = C, C is self-dual. A binary self-dual code is doubly-even if all codewords have weight divisible by four, and singly-even if there is at least one nonzero codeword of weight 2 (mod 4). Self-dual doubly-even codes exist only if n is a multiple of eight. The weight enumerator W (y) of a code C is defined as W (y) = n i=0 A iy i, where A i is the number of codewords of weight i in C. We say that two binary linear codes C and C are equivalent if there is a permutation of coordinates which sends C to C. The set of coordinate permutations that maps a code C to itself forms a group called the automorphism group of C (denoted by Aut(C)). Let S n be the symmetric group of degree n. We say that a permutation σ S n is of type p (c, f) if it has exactly c cycles of length p and f fixed point in its decomposition. A duo is a set of two coordinate positions of a code. A cluster is a set of disjoint duos such that any union of any two duos is the support of a vector of weight 4 in the code. A d-set for a cluster is a subset of coordinates such that there is precisely one element of each duo in the d-set. A defining set for a code will consist of a cluster and a d-set provided the code is generated by the weight-4 vectors arising from the cluster and the vector whose support is the d-set. A t (v, k, λ) design D is a set X of v points together with a collection of k-subsets of X (named blocks) such that every t-subset of X is contained exactly in λ blocks. The block intersection numbers of D are the cardinalities of the intersections of any two distinct blocks. A t (v, k, λ) design is called self-orthogonal if the block intersection numbers have the same parity as the block size k. It was shown in [17] that the minimum weight d of a binary self-dual code of length n is bounded by { 4 n d , if n 22 (mod 24); 4 n , if n 22 (mod 24). (1) We call a self-dual code meeting this upper bound extremal. A self-dual code which has the largest minimum weight among all self-dual codes of a given length is named optimal. All optimal codes of length 42 and 44 having an automorphism of order 5 are classified up to equivalence in [3] and [2]. The extremal binary self-dual [46, 23, 10] code is unique (see [12, Lemma 2.2]). A complete classification of binary self-dual [48, 24, 10] codes with an automorphism of odd prime order is given in [7]. Recently, all optimal binary self-dual codes of lengths 52 to 60 having an automorphism of order 7 or 13 were classified in [19]. So we have been intrigued to investigate optimal self-dual codes of length n 50 possessing an automorphism of order 5. From [14, Table 3] we have the following cases for the length n and the type of automorphism: n = 50, type 5 (8, 10);

3 New binary self-dual codes of lengths 50 to 60 3 n = t, type 5 (10, 2t), t = 0, 1,..., 6. We can exclude the case of a self-dual [62, 31, 12] 2 code with an automorphism of type 5 (10, 12) due to a result from [5]. In this paper we give a full classification of all optimal binary self-dual codes of lengths 50 n 60 with an automorphism of order 5. To do so we continue with some properties of the binary self-dual codes having an automorphism of prime order. 2 Construction Method Let C be a binary self-dual code of length n with an automorphism σ = (1, 2,..., p)(p + 1, p + 2,..., 2p) (p(c 1) + 1, p(c 1) + 2,..., pc), (2) of type p (c, f), where f = n pc. Denote the cycles of σ by Ω 1, Ω 2,..., Ω c, and the fixed points by Ω c+1,..., Ω c+f. Let F σ (C) = {v C vσ = v}, E σ (C) = {v C wt(v Ω i ) 0 (mod 2), i = 1,..., c + f}, where v Ω i is the restriction of v on Ω i. Theorem 1 [13] Assume C is a self-dual code having an automorphism of type p (c, f). The code C is a direct sum of the subcodes F σ (C) and E σ (C). F σ (C) and E σ (C) are subspaces of dimensions c+f 2 and (p 1)c 2, respectively. From the definition of F σ (C) it follows that v F σ (C) iff v C and v is constant on each cycle. Let π : F σ (C) F c+f 2 be the projection map where if v F σ (C), (vπ) i = v j for some j Ω i, i = 1, 2,..., c + f. Denote by E σ (C) the code E σ (C) with the last f coordinates deleted. So E σ (C) is a self-orthogonal binary code of length pc. For v in E σ (C) we let v Ω i = (v 0, v 1,..., v p 1 ) correspond to the polynomial v 0 +v 1 x+ +v p 1 x p 1 from P, where P is the set of even-weight polynomials in F 2 [x]/ x p 1. Thus we obtain the map ϕ : E σ (C) P c. Theorem 2 [20] A binary [n, n/2] code C with an automorphism σ defined in (2) is self-dual if and only if the following two conditions hold: (i) C π = π(f σ (C)) is a binary self-dual code of length c + f, (ii) for every two vectors u, v C ϕ = ϕ(e σ (C) ) we have c i=1 u i (x)v i (x 1 ) = 0. If 2 is a primitive root modulo p then C ϕ is a self-dual code of length c over the field P = F 2 p 1 under the inner product (u, v) = c u i vi 2(p 1)/2. To classify all codes, we need additional conditions for equivalence and we use the following theorem. Theorem 3 [21] The following transformations preserve the decomposition and send the code C to an equivalent one: (i) a permutation of the fixed coordinates; (ii) a permutation of the p-cycles coordinates; (iii) a substitution x x 2 in C ϕ and (iv) a multiplication of the j-th coordinate of C ϕ by x tj where t j is an integer, 0 t j p 1, j = 1, 2,..., c. i=1

4 4 Nikolay Yankov, Moon Ho Lee Table 1 The multiplicative group of the field P = F 16 e α α α α α α α α α α α α α α Since 2 is a primitive root modulo 5, using Theorem 2, the subcode C ϕ is a self-dual code of length c over the field P under the inner product (u, v) = c u i vi 4. (3) Furthermore P is a finite field with 16 elements, P = F 16 = {0, e = α 0, α k k = 1,..., 14}, where e = x + x 2 + x 3 + x 4, α = 1 + x is a primitive element of multiplicative order 15. We list the elements of P the multiplicative group of P in Table 1. Denoting δ = α 5 the group P can also be described as P = {α 3t δ l 0 t 4, 0 l 2}. i=1 3 Binary self-dual [50, 25, 10] codes with an automorphism of type 5 (8, 10) Let C be a putative binary [50, 25, 10] self-dual code having an automorphism of order 5 with 8 cycles and 10 fixed points. According to Theorem 2 we have that C π is a binary self-dual [18, 9] code. There are exactly 9 such codes (see [16]). Using the notation from [15] these codes are: 9i 2, 5i 2 e 8, 3i 2 d 12, 2i 2 2e 7, i 2 2e 8, i 2 2d 8, i 2 d 16, 3d 6, and d 10 e 7 f 1. Let X c be the set of the cycle positions and X f the set of the fixed positions in the binary [18, 9] code C π. Then X c X f = {1,..., 18}, X c X f =. Since we are looking for a [50, 25] code with minimum distance 10 the next statement is obvious. Lemma 1 Every vector in C π with weight 2 or 4 must have at least 2 elements of its support in X c. Proposition 1 There does not exist a binary self-dual [50, 25, 10] code with an automorphism of type 5 (8, 10). Proof Because we have 8 cycle coordinates the codes 9i 2 and 5i 2 e 8 obviously do not satisfy Lemma 1. In the case 3i 2 d 12 six cycle coordinates should be the support of the 2-weight vectors from 3i 2. Considering the cluster {{1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}, {11, 12}} for d 12 we must have at least six cycle coordinates from that cluster which is obviously impossible. In the case C π = 2i2 2e 7 according to Lemma 1 we can fix four cycle coordinates to be the support of the 2i 2 summand. The code 2e 7 has two clusters Q 1 = {{1, 2}, {3, 4}, {5, 6}}, Q 2 = {{8, 9}, {10, 11}, {12, 13}}, and

5 New binary self-dual codes of lengths 50 to 60 5 two d-sets, d 1 = {1, 3, 5, 7}, d 2 = {8, 10, 12, 14}, that form a defining set. We must take at least four coordinates from {1,..., 7} in X c. The same is true for the set {8,..., 14}, corresponding to the other e 7 summand a contradiction since X c = 8 < 12. An observation of codewords of weights 2 and 4 and similar arguments as above, show that no arrangement of X c and X f for the remaining 4 codes could satisfy Lemma 1. 4 Codes with an automorphism of type 5 (10, 2t) Let C be an optimal binary self-dual code having an automorphism of order 5 with 10 cycles and f = 2t, t = 0,..., 5 fixed points. Proposition 2 Let C ϕ be a [10, 5] code over P, self-dual under the orthogonality condition (3), such that E σ (C) is a code with minimum distance at least 10. Then the generator matrix of C ϕ is e a 16 a 17 a 18 a 19 a 1,10 0 e a 26 a 27 a 28 a 29 a 2,10 G ϕ = 0 0 e 0 0 a 36 a 37 a 38 a 39 a 3, e 0 a 46 a 47 a 48 a 49 a 4,10, (4) e a 56 a 57 a 58 a 59 a 5,10 for a j6 {0, e, δ, δ 2 }, j = 1,..., 5. Furthermore (a 16, a 17, a 18, a 19, a 1,10 ) is one of the following vectors (0, e, e, δ, δ 2 ), (e, e, e, e, e), (e, δ, δ, δ, δ), (e, δ, δ, δ 2, δ 2 ), (e, e, e, δ, δ). Proof Let G ϕ be in a row reduced form. Using transformation (iv) from Theorem 3 we can assume that the elements in the first row of G ϕ are from the set {0, e, δ, δ 2 }. Further interchanging the columns of G ϕ, it follows that, we can take 0 a 16 a 17 a 18 a 19 a 1,10 δ 2. Using (3) we reduce the vector v = (a 16, a 17, a 18, a 19, a 1,10 ) to cases listed in Table 2. The transformation γ : x x 2, (iii) from Theorem 3, maps δ to δ 2 and vice versa. Under γ the vectors v 4, v 8, and v 11 correspond to v 3, v 7, and v 10, respectively. Using ϕ 1, every element in G ϕ corresponds to a 4 5 circulant matrix. Denote the rows of the generator matrix of E σ (C) = ϕ 1 (G ϕ ) by r i, i = 1,..., 20. If case of v 1, we have wt(r 1 ) = 8. Further for v 2 we have that r 1 + r 2 = ϕ 1 (α, 0, 0, 0, 0, 0, 0, α, α, α) is a vector of weight 8. Lastly for v 3 = (0, 0, e, δ, δ), it follows that the weight of r 1 + r 3 = ϕ 1 (α 2, 0, 0, 0, 0, 0, 0, α 2, α 7, α 7 ) is again 8. Since P = {α 3t δ l }, 0 t 4, 0 l 2 every element a j,6 P, j = 2,..., 5 can be transformed into e, δ or δ 2 using a multiplication of j-th row of G ϕ by α 3t, followed by some cyclic shifts in the j-th column. A computer program for calculating all codes with generator matrix of the form (4), using the orthogonality condition, was created. It turns out that there are exactly 56 inequivalent (under the maps from Theorem 3) [10, 5] codes satisfying Proposition 2. Actually, all codes can be obtained using just

6 6 Nikolay Yankov, Moon Ho Lee Table 2 Cases for the first row of G ϕ v 1 = (0, 0, 0, 0, e) v 2 = (0, 0, e, e, e) v 3 = (0, 0, e, δ, δ) v 4 = (0, 0, e, δ 2, δ 2 ) v 5 = (0, e, e, δ, δ 2 ) v 6 = (e, e, e, e, e) v 7 = (e, δ, δ, δ, δ) v 8 = (e, δ 2, δ 2, δ 2, δ 2 ) v 9 = (e, δ, δ, δ 2, δ 2 ) v 10 = (e, e, e, δ, δ) v 11 = (e, e, e, δ 2, δ 2 ) Table 3 Generators elements of H i, i = 1,..., 56 a 26,..., a 5,10 a 26,..., a 5,10 a 26,..., a 5,10 H ef08d540953e H ef fc7 H ef bf1 H efa02e1a5d38 H d751a4f52b9b H b080fe46 H a1c82ae3f6 H a2438aa1d9 H e4f H af3c5e4c7 H a137dae2d9 H a20e1aabb0 H fb03ebc0fc15 H fb51bd H fb5a14f5ef5d H fba182caec4b H fba2ed9aa055 H a5d38ab2a3 H aaa56f6ab3e1 H d8d51ea95a4f1 H d8d529005ef5d H d8da572cab867 H b c0e6 H b08a0a55a84b0 H b08acb9fae8ca H ebc507845e4c7 H ebca02e1a8eca H ebcac182aea0e H f54a0ccaa812c H bc ee6 H bc fc7 H aa54b6a5f6d3 H aaa084babc9f H ce087fc09470 H cea0c6da59e4 H eb082aa0d714 H da13beae201 H da202aaab32 H a2e01aa055 H aa51b8f5a8dc H d3f529005ef24 H bceacbe4ae86d H e95a022aa8e6d H be14a42dfaae8c H d00a fb1 H d00c2e02ba4067a3 H d01725a1f87aed86 H d01ae808ea10d610 H d01c0aa15b6aea0e H d0e1b4a1f87aaa5f H ac01ac05a7235ef42 H bf915053c51405 H cf0d2d b5f7 H d530b959d545b2c5 H c554105c503 H cadd0aaf5dd Table 4 Size of the automorphism groups of optimal codes over F 16 Aut(C) # one vector for the first row v 7 = (e, δ, δ, δ, δ). Denoting the generator matrices by H i, i = 1,..., 56, we list the generating parameters for all obtained codes in Table 3. Remark 1 The notation in Table 3 is hexadecimal: t = 0,..., e denote α t, whereas f denotes the zero of P. The size of the automorphism groups and the weight enumerators of the pre-images ϕ 1 are listed in Tables 4 and 5, respectively. Since the minimum weight of all codes is 12, they can be used for constructing binary self-dual codes of lengths 50 n 60 with an automorphism of type 5 (10, f), f = 2l, l = 0,..., [50, 25, 10] codes with an automorphism of order 5 For a binary self-dual [50, 25, 10] code there are two possibilities for the weight enumerator (see [14]): W 50,1 = y , 368y , 752y 14 +,

7 New binary self-dual codes of lengths 50 to 60 7 Table 5 Enumerators of the self-orthogonal [50, 20, 12] codes generated by the matrix ϕ 1 (H i ), i = 1,..., 56 # A 12 A 16 A 20 A 24 A 28 A 32 A 36 A W 50,2 = 1 + (580 32β)y 10 + ( β)y 12 +, where 0 β 2 [11]. Codes exist both for W 50,1 and for W 50,2 when β = 0, 1, 2 (see [14] and references therein). Let C be a binary self-dual [50, 25, 10] code, possessing an automorphism of type 5 (10, 0). According to Theorem 2 the subcode C π is a binary selfdual [10, 5, 2] code. There are two such codes, namely 5i 2 and i 2 e 8 [15] with generator matrices B 1 = and B 2 = , respectively. We have constructed the two direct summands for the code C and next we have to attach them together. Let the subcode E σ (C) be fixed as generated by H j, j = 1,..., 56. We need to consider all equivalent possibilities for the second subcode F σ (C) with generator matrix B i, i = 1, 2. Denote by G i, i = 1, 2 the automorphism group of the code generated by B i, i = 1, 2. We have that G 1, generated by (1, 8, 4)(2, 10, 7, 5)(3, 9, 6) and (1, 9, 10, 8, 2, 6, 4, 5, 3, 7), is a group of cardinality The other group G 2 = (2, 7, 8, 3, 10, 5)(4, 9), (1, 6)(2, 7, 9, 4, 3, 8, 10) has cardinality Let

8 8 Nikolay Yankov, Moon Ho Lee Table 6 Size of the automorphism groups of [50, 25, 12] codes Aut(C) enumerator # W 50,2, β = W 50,2, β = τ S 10 be a permutation. Denote by C50,i,j τ, i = 1, 2, j = 1,..., 56 the [50, 25] self-dual code determined by H j as a generator for E σ (C) and the matrix B i with columns permuted by τ as a generator matrix for F σ (C). It is easy to see that if τ 1 and τ 2 belong to one and the same right coset of S 10 to G i, then the codes C τ1 50,i,j and Cτ2 50,i,j are equivalent. We need to consider only the permutations τ S 10 from the right transversal R i, i = 1, 2 of S 10 with respect to G i, i = 1, 2. After calculating all codes C50,i,j τ, i = 1, 2, j = 1,..., 56, τ R i we have the following. Proposition 3 There are exactly 270 inequivalent self-dual [50, 25, 10] binary codes having an automorphism of type 5 (10, 0). 147 codes possess weight enumerator W 50,2 for β = 0 and 123 have weight enumerator W 50,2 for β = 2. Remark 2 We present the cardinality of the automorphism groups of all obtained codes in Table 6. The two codes with an automorphism group of size 50 constructed here are known. They are the pure double-circulant self-dual codes P 50,1 and P 50,2 from [10, Table 2]. All other 268 codes are new. 4.2 [52, 26, 10] codes with an automorphism of order 5 For a binary self-dual [52, 26, 10] code there are two possible forms for the weight enumerator [14]: W 52,1 = y y , 800y 14 +, (5) W 52,2 = 1 + (442 16β)y 10 + ( β)y 12 +, (6) where 0 β 12, β 11 [4]. Codes exist with W 52,1 and with W 52,2 for all values of the parameter except β = 10 [18]. Similar to Section 4.1 we have the following. Proposition 4 There are exactly inequivalent binary [52, 26, 10] selfdual codes having an automorphism of type 5 (10, 2). One of these codes has weight enumerator W 52,2 for β = 10. Theorem 4 There exists an optimal [52, 26, 10] binary self-dual code with weight enumerator W if and only if W = W 52,2 in (6) with β {0,..., 12}, β 11 or W is given by (5).

9 New binary self-dual codes of lengths 50 to 60 9 Table 7 Enumerators and automorphism groups of [52, 26, 10] codes Aut(C) enumerator # W 52, W 52,2, β = W 52,2, β = W 52,2, β = W 52,2, β = W 52,2, β = Remark 3 The code with the new value β = 10 is generated by the matrices H 42 and C π = We list the cardinality of the automorphism groups and the type of weight enumerators of all constructed codes in Table 7. The two codes with an automorphism group of size 50 are known. They are the two bordered double-circulant codes B 52,1 and B 52,2 (see [10, Table 4]). 4.3 [54, 27, 10] codes with an automorphism of order 5 There are two possible forms for the weight enumerator of a binary self-dual [54, 27, 10] code [14]: W 54,1 = 1 + (351 8β)y 10 + ( β)y 12 +, 0 β 43, W 54,2 = 1 + (351 8β)y 10 + ( β)y 12 +, 12 β 43. Codes exist with W 54,1 when 0 β 19, 26 and for W 54,2 when 12 β 21, 24, 26, and 27 (see [6,14,19]). Using similar arguments like in Section 4.1 we have the following. Proposition 5 Up to equivalence, there are exactly inequivalent binary self-dual [54, 27, 10] codes having an automorphism of type 5 (10, 4). There exist binary self-dual [54, 27, 10] codes with weight enumerator W 54,1 for β = 20, 22 and with W 52,2 for β = 22. Remark 4 We list the obtained codes by their weight enumerators in Table 8. The new values β = 20, 22 in W 54,1 and β = 22 in W 54,2 are marked with bold font. Examples for codes with the new weight enumerators are: W 54,1, β = 20: C (1,7,6,10)(2,8,5,9,4) 54,4,6 ; W 54,1, β = 22: C (2,10,8,3)(4,9,7,6,5) 54,6,7 ; W 54,2, β = 22: C (2,4)(3,8,9,7,6,5) 54,5,2.

10 10 Nikolay Yankov, Moon Ho Lee Table 8 Weight enumerators of [54, 27, 10] codes β = 0 β = 5 β = 10 β = 15 β = 20 total C β = 12 β = 17 β = 22 β = 27 total C β = 2 β = 7 β = 12 β = 17 β = 22 total C Here for a permutation τ S 10, denote by C54,i,j τ, i = 4, 5, 6, j = 1,..., 56 the [54, 27] self-dual code determined by H j as a generator for E σ (C) and the matrix C i with columns permuted by τ as a generator matrix for F σ (C). The three matrices , , , are denoted by C 4, C 5, and C 6, respectively. 4.4 Singly- and doubly-even [56, 28, 12] codes with an automorphism of order 5 For a [56, 28, 12] binary self-dual codes there is one possible form for the weight enumerator for a doubly-even code W 56,1 = y y (7) Codes are known for this weight enumerator (see [9]). There is also one possibility for a singly-even code W 56,2 = y y y (8) No codes are known for this weight enumerator. Again using methods similar to Section 4.1 we conclude. Proposition 6 There does not exist a singly-even self-dual [56, 28, 12] code possessing an automorphism of order 5. There are exactly 3763 inequivalent doubly-even self-dual [56, 28, 12] codes having an automorphism of order 5. Using previous results (see [19, Corollary 1] and [14, Table 3]) we have the following. Corollary 1 There does not exist a singly-even self-dual [56, 28, 12] code with an automorphism of odd prime order p > 3.

11 New binary self-dual codes of lengths 50 to In [9] exactly 1151 inequivalent doubly-even [56, 28, 12] codes are constructed. One of these codes has an automorphism of order 7 and none of order 5. There are also 4202 inequivalent binary doubly-even [56, 28, 12] selfdual codes having an automorphism of type 7 (8, 0) (see [19]). None of these codes have an automorphism of order 5 and we can state. Proposition 7 There exist at least 9115 binary doubly-even [56, 28, 12] selfdual codes, up to equivalence. Harada [9] proved that any binary doubly-even [56, 28, 12] self-dual code generates a self-orthogonal 3 (56, 12, 65) design with block intersection numbers 0, 2, 4, and 6. Also, two inequivalent extremal doubly-even self-dual codes of length 56 give two non-isomorphic self-orthogonal 3 (56, 12, 65) designs. So we have the following. Corollary 2 There are at least 9115 inequivalent 3 (56, 12, 65) self-orthogonal designs with block intersection numbers 0, 2, 4, [58, 29, 10] codes with an automorphism of order 5 There are two possible forms for the weight enumerator of a putative binary self-dual [58, 29, 10] code: W 58,1 = 1 + (165 2β)y 10 + ( β)y 12 +, 0 β 82, W 58,2 = 1 + (319 24β 2γ)y 10 +, where 0 β 11 and 0 γ β. A code for W 58,1 exists when β = 55 [14]. Codes exist (see [14,19] ) for W 58,2 when: β = 0 with γ = 0, 2, 16, 18, 20, 30, 32, 142, 158, and γ {2m 18 m 64}; β = 1 with γ {2m 21 m 50}; β = 2 with γ =32, 36, 40, 44, and γ {2m 24 m 44} {92}. Let C be a binary self-dual [58, 29, 10] code, possessing an automorphism of type 5 (10, 8); thus the subcode C π is a binary self-dual [18, 9, 2] code. There are 9 such codes: 9i 2, 5i 2 e 8, 3i 2 d 12, 2i 2 2e 7, i 2 2e 8, i 2 2d 8, i 2 d 16, 3d 6, and d 10 e 7 f 1 [15]. Since the code C has minimum distance 10, Lemma 1 is valid and using it we can exclude some of the codes. Thus only the codes i 2 2d 8, i 2 d 16, 3d 6, and d 10 e 7 f 1 remain, with generator matrices B i = (I 9 M i ), i = 6,..., 9, where M 6 = M 7 = M 8 = M 9 =

12 12 Nikolay Yankov, Moon Ho Lee Table 9 Generator matrices for C π in [58, 29, 10] codes name base matrix permutation C 10 B 6 (3, 11, 12, 13, 14, 15, 16, 17, 10, 9, 8, 7, 6, 5, 4) C 11 B 7 (4, 11, 9, 12, 10, 8, 7, 6, 5) C 12 B 7 (4, 11, 9, 12, 13, 10, 8, 7, 6, 5) C 13 B 8 (10, 11, 12) C 14 B 8 (10, 11, 12, 13, 14, 15, 16, 17) C 15 B 8 (8, 11, 13, 15, 17, 18, 10, 12, 14, 16, 9) C 16 B 9 (9, 11, 13)(101214) C 17 B 9 (4, 11, 13, 14, 10, 12, 9, 8, 7, 6, 5) C 18 B 9 (4, 11, 14, 9, 12, 8, 7, 6, 5)(10, 13, 15, 16) After calculating all possible ( 18 8 ) choices for the fixed positions Xf, we have 9 possible codes with generator matrices listed in Table 9. Similar to Section 4.1 we have the following. Proposition 8 There are exactly inequivalent singly-even self-dual [58, 29, 10] codes possessing an automorphism of type 5 (10, 8). There exist binary self-dual [58, 29, 10] codes with weight enumerator W 58,2 for β = 0, γ = 10, 12, 16, 22, 26, 30, 130, and 136; β = 1, γ = 26, 36, 106, 116, and 126; β = 2, γ = 0, 100, and 110. All codes obtained from the matrices C 10,..., C 16 have weight enumerator W 58,2 for different values of β and γ listed in Table 10. All new obtained values for the parameters are listed in bold font. The codes with parameters listed in Proposition 8 are the first examples of codes with their respective weight enumerator. The matrices C 17 and C 18 generate codes with weight enumerator W 58,1 for β = 55. We conclude this subsection with Table 12 that gives examples for every new value of (β, γ) in W 58,2 obtained. 4.6 [60, 30, 12] codes with an automorphism of order 5 There are two possible forms for the weight enumerator of a binary self-dual [60, 30, 12] code: W 60,1 = y y y 16 +, W 60,2 = 1 + ( β)y 12 + ( β)y 14 +, 0 β 10. A code exists for W 60,1 [8] and for W 60,2 when β = 0, 1, 7, and 10 [14]. Similar to Section 4.1 we have calculated all codes. We summarize the results in the following. Proposition 9 There are exactly 79 inequivalent singly-even self-dual binary [60, 30, 12] codes possessing an automorphism of type 5 (10, 10). There exist at least two inequivalent self-dual [60, 30, 12] codes with weight enumerator W 60,2 for β = 5.

13 New binary self-dual codes of lengths 50 to Table 10 Weight enumerator of the [58, 29, 10] codes generated by C 10,..., C 16 C 10 # β γ C 14 # β γ C 11 # β γ C 15 # β γ C 12 # β γ C 16 # β γ C 13 # β γ Table 11 Order of the automorphism groups of the constructed [58, 29, 10] codes case Aut C 10 C 11 C 12 C 13 C 14 C 15 C 16 C 17 C 18 total Remark 5 In this paper for computing the automorphism groups of the codes and also checking for code equivalence we use the computer system Q-extensions by Iliya Bouyukliev [1].

14 14 Nikolay Yankov, Moon Ho Lee Table 12 Generator parameters for new [58, 29, 10] codes β γ C π τ C ϕ 0 10 C 14 (1, 8)(2, 7, 3, 4, 6)(5, 9, 10) H C 11 (1, 5, 3, 4, 2)(9, 10) H C 13 (1, 2, 7, 8)(9, 10) H C 11 (3, 4, 5, 6, 7, 9, 8) H C 13 (4, 7, 9, 6)(5, 8) H C 14 (4, 5, 10, 9, 7, 6) H C 14 (1, 6, 8)(2, 9)(5, 10, 7) H C 13 (1, 2, 3)(4, 5, 10, 7, 6) H C 15 (1, 4, 6, 3, 2, 5, 8, 9)(7, 10) H C 15 (2, 4, 3)(8, 10, 9) H C 15 (2, 4, 7, 6, 3)(5, 10) H C 15 (1, 2, 6, 5, 4, 3, 10, 8, 9) H C 15 (2, 7, 4, 5, 6) H C 10 (1, 2) H C 16 (1, 4, 10)(2, 3, 5)(7, 9, 8) H C 16 (1, 2, 5)(3, 6)(7, 9, 8) H 3 Acknowledgment We thank the anonymous referees for the helpful comments and suggestions, which significantly contributed to improving the quality of the publication. This paper was studied with the support of the Ministry of Education Science and Technology (MEST) and the Korean Federation of Science and Technology Societies (KOFST). This work was also supported by World Class University Project (WCU) R , Basic Science Research Program , NRF Korea, and MEST , NRF Korea. The first author was also supported by Shumen University under Grant RD / References 1. I. Bouyukliev, About the code equivalence, Advances in Coding Theory and Cryptography. Series on coding theory and cryptology, vol. 3. World Scientific Publishing, (2007) 2. S. Bouyuklieva, New extremal self-dual codes of lengths 42 and 44. IEEE Trans. Inform. Th. 43, (1997) 3. S. Bouyuklieva, Some optimal self-orthogonal and self-dual codes. Discrete Math. 287, 1 10 (2004) 4. S. Bouyuklieva, M. Harada, A. Munemasa, Restrictions on the weight enumerators of binary self-dual codes of length 4m. In: Proc. International Workshop Optimal Codes and Related Topics, (OCRT 07), pp White Lagoon, Bulgaria (2007) 5. S. Bouyuklieva, A. Malevich, W. Willems, Automorphisms of extremal self-dual codes. IEEE Trans. Inform. Th. 56, (2010) 6. S. Bouyuklieva, P. Östergård, New constructions of optimal self-dual binary codes of length 54. Designs, Codes and Cryptography 41, (2006) 7. Bouyuklieva, S., Yankov, N., Kim, J.L.: Classification of binary self-dual [48, 24, 10] codes with an automorphism of odd prime order. Finite Fields and Their Applications 18(6), (2012). 8. J. H. Conway, N.J.A. Sloane, A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inform. Th. 36, (1990)

15 New binary self-dual codes of lengths 50 to M. Harada, Self-orthogonal 3 (56, 12, 65) designs and extremal doubly-even self-dual codes of length 56. Designs, Codes and Cryptography 38, 5 16 (2006) 10. M. Harada, T.A. Gulliver, H. Kaneta, Classification of extremal double circulant selfdual codes of length up to 62. Discrete Math. 188, (1998) 11. M. Harada, A. Munemasa, Some restrictions on weight enumerators of singly even selfdual codes. IEEE Trans. Inform. Th. 52, (2006) 12. M. Harada, A. Munemasa, V. Tonchev, A characterization of designs related to an extremal doubly-even self-dual code of length 48. Ann. Combin 9, (2005) 13. W.C. Huffman, Automorphisms of codes with application to extremal doubly-even codes of length 48. IEEE Trans. Inform. Th. 28, (1982) 14. W.C. Huffman, On the classification and enumeration of self-dual codes. Finite Fields Appl. 11, (2005) 15. W.C. Huffman, V. Pless, Fundamentals of error correcting codes. Cambridge University Press (2003) 16. V. Pless, A classification of self-orthogonal codes over GF(2). Discrete Math. 3, (1972) 17. E.M. Rains, Shadow bounds for self-dual codes. IEEE Trans. Inform. Th. 44, (1998) 18. N. Yankov, New optimal [52, 26, 10] self-dual codes. to appear in Designs, Codes and Cryptography (2012) 19. N. Yankov, R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13. IEEE Trans. Inform. Th. 56, (2011) 20. V. Yorgov, Binary self-dual codes with an automorphism of odd order. Problems. Info. Transmission 19, (1983) 21. V. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56. IEEE Trans. Inform. Th. 33, (1987)

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