New Extremal Self-Dual Codes of Length 68 Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA and Masaaki Harada Department of Mathematical Sciences Yamagata University Yamagata 990–8560, Japan June 22, 2011

1

Running head: New Extremal Self-Dual Codes of Length 68 Name: Steven T. Dougherty and Masaaki Harada Contact Author: Steven T. Dougherty Address: Department of Mathematics University of Scranton Scranton, PA 18510, USA Telephone: 717-941-6104 Fax: 717-941-5981 E-mail: [email protected]

2

Abstract In this correspondence, we give two computational results on binary self-dual codes. A number of extremal singly-even self-dual [68, 34, 12] codes with weight enumerators not known to exist, are constructed. For k ≤ 10, the relationship between extremal doubly-even self-dual codes of length 24k and certain singly-even self-dual codes of length 24k − 2 is presented.

Index Term – New extremal doubly-even codes, shadow codes, weight enumerators.

3

1

Introduction

A linear binary [n, k, d] code is a k-dimensional subspace of F2n with minimum weight d. A non-linear code is simply a set of vectors. In this correspondence, we consider only binary codes, specifically self-dual codes, namely those codes C for which C = C ⊥ , where C ⊥ denotes the orthogonal under the standard inner-product. An extremal self-dual code is a code having the largest minimum weight for the given length. If a self-dual code has only vectors of doubly-even weight then we say that the code is a doubly-even code, otherwise we say that it is a singly-even code. We denote the weight enumerator by: X WC (x, y) = Ai xn−i y i where there are Ai vectors of weight i in C. When recording a weight enumerator we shall set x = 1, and when it is symmetric we will only show half. Conway and Sloane [2] defined the shadow of a self-dual code. The shadow provide restrictions on the weight enumerators of binary extremal self-dual codes, and was used to determine new upper bounds for the minimum weight of binary self-dual codes in [2]. In Section 2, a number of extremal singly-even self-dual [68, 34, 12] codes with weight enumerators not known to exist, are constructed, giving the possible weight enumerators of such codes. In Section 3, the relationship between extremal doubly-even self-dual codes of length 24k and certain singly-even self-dual codes of length 24k − 2 is presented for k ≤ 10. This relationship provides a condition for the existence of an extremal doubly-even code of length 24k. The first code in this class is the extended [24, 12, 8] Golay code. The second code is an extremal doubly-even [48, 24, 12] code. Only one extremal doubly-even code is known of length 48 which is the extended quadratic residue code. It is not known if there exists other extremal doubly-even codes of length 24k. In particular, the existence of an extremal doubly-even self-dual code of length 72 is a long-standing open question (cf. [7]).

2

Extremal Singly-Even [68, 34, 12] Codes

Applying Theorem 5 in [2] to a putative self-dual [68, 34, 14] code and examining the weight enumerator of the shadow shows that there is no possible self-dual code for these parameters. Applying the same to a [68, 34, 12] code, the possibilities for the weight enumerators of such a code and its shadow are determined. The weight enumerators are given in Tables 1 and 2. In these tables, a6 , a7 and a8 are undetermined parameters in Theorem 5 of [2]. It is easy to see that a6 has to be a multiple of 4 and so a6 = 4β. Since the number of vectors of weight 2 in the shadow is either 0 or 1 then a8 = 0 or 16384. If there is 1 vector of weight 2 in the shadow then we have the number of weight 6 vectors in the shadow is 0.

4

Table 1: Possible Weight Enumerators for [68, 34, 12] Codes the coefficient of y i 1 442 + a6 14960 − 2a6 + a7 −9a6 + a8 + 174471 − 8a7 1478048 − 14a8 + 20a6 + 22a7 35a6 + 89a8 − 8a7 + 9546537 −90a6 − 336a8 − 83a7 + 46699952 820a8 + 160a7 − 75a6 + 175078410 8a7 + 509477760 + 240a6 − 1288a8 −352a7 + 90a6 + 1092a8 + 1160564636 338a7 − 420a6 + 2081169376 + 208a8 −42a6 − 2002a8 + 208a7 + 2949602799 3312254400 + 504a6 − 572a7 + 2860a8

weight i 0 12, 56 14, 54 16, 52 18, 50 20, 48 22, 46 24, 44 26, 42 28, 40 30, 38 32, 36 34

Table 2: Possible Weight Enumerators for the Shadows of [68, 34, 12] Codes the coefficient of y i 1 a 16384 8 −1 1 a − a 256 7 1024 8 7 15 a + 2048 a8 + 14 a6 128 7 −91 35 a − 1024 a8 + 29920 − 3a6 256 7 33 455 a + 4096 a8 + 2956096 + 91 a 2 6 64 7 273 −55a6 + 93399904 − 1024 a8 − 1001 a 256 7 1001 1001 495 a + 1018955520 + a + a6 8 7 2048 128 4 −3003 715 a7 + 4162338752 − 198a6 − 1024 a8 256 429 a + 231a6 + 6435 a + 6624508800 32 7 8192 8

weight i 2, 66 6, 62 10, 58 14, 54 18, 50 22, 46 26, 40 30, 38 34

It follows that in this case a7 = −4096 and we obtain the weight enumerator W1 = 1 + (442 + 4β)y 12 + (10864 − 8β)y 14 + · · · where β is an undetermined parameter. Then the weight enumerator of the shadow is S1 = y 2 + (β − 104)y 10 + (30816 − 12β)y 14 + · · · . If there are no vectors of weight 2 in the shadow then the number of weight 6 vectors 1 in the shadow is − 256 a7 = γ and therefore a7 = −256γ. In this case we obtain the weight enumerator W2 = 1 + (442 + 4β)y 12 + (14960 − 8β − 256γ)y 14 + · · · where β and γ are undetermined parameters. Then the weight enumerator of the shadow is S2 = γy 6 + (β − 14γ)y 10 + (91γ − 12β + 29920)y 14 + · · · . It was shown in [4] that there exist exactly 107 inequivalent extremal double circulant self-dual [68, 34, 12] codes with 17 different weight enumerators The values β of their weight 5

enumerators are given in Table 3, where the double circulant codes with W2 have γ = 0. We remark that two codes were found by computer by Neil Sloane and communicated in a private communication. Their weight enumerators correspond to W2 with β = 238 and γ = 0 and W1 with β = 203. Table 3: Weight Enumerators of Double Circulant [68, 34, 12] Codes W1 W2

104, 137, 170, 203, 236, 269, 302, 335, 401 34, 68, 102, 136, 170, 204, 238, 272

Proposition 2.1 (Harada [5]) Let Ω be a subset of the set {1, 2, · · · , n} such that |Ω| is odd if 2n ≡ 0 (mod 4) and |Ω| is even if 2n ≡ 2 (mod 4). Let G0 = ( In , A ) be a generator matrix of a self-dual code C0 of length 2n where In is the identity matrix of order n. Then the following matrix 

G=

     



1 0 x1 · · · xn 1 · · · 1   y1 y1   .. ..  . . In A  yn yn

where xi = 1 if i ∈ Ω and xi = 0 otherwise, and yi = xi + 1 (1 ≤ i ≤ n), generates a self-dual code C of length 2n + 2. Let C1 and C2 be self-dual codes of length 66 with generator matrices ( I33 , R ), where R are circulant matrices with first rows (001010100111000110100111100000000) and (010011011111010011011010011010000), respectively. These codes are extremal double circulant singly-even codes. Applying Proposition 2.1 to these codes, we have found extremal singly-even [68, 34, 12] codes having weight enumerators for which extremal codes were not previously known to exist. In Table 4, we list Ω and C0 given in Proposition 2.1 together with the values β and γ of the weight enumerators W . Recently other extremal self-dual codes of length 68 have been constructed also in [1]. Note that the results in this paper are listed in [1]. Notice that extremal singly-even [68, 34, 12] codes are abundant and many have different weight enumerators. However, the shadow optimal [68, 34, 12] code (that is, a6 = a7 = a8 = 0) remains very elusive (see [3] for the definition of shadow optimal codes), since the code is closely related to an elusive extremal doubly-even code of length 72.

6

Table 4: New Extremal Singly-Even [68, 34, 12] Codes Codes C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34 C35 C36 C37 C38 C39 C40 C41 C42 C43 C44 C45 C46 C47 C48

Ω 1, 2 1, 3 1, 5 1, 6 1, 7 1, 8 1, 9 1, 10 1, 13 1, 14 1, 17 1, 4 1, 12 1, 2, 4, 5, 8, 20 1, 2, 15, 25, 28, 31 1, 3, 5, 14, 16, 29 1, 3, 20, 24, 27, 30 1, 4, 7, 11, 17, 21 1, 4, 7, 11, 24, 30 1, 4, 8, 14, 17, 22 1, 2, 8, 10, 14, 20, 27, 30 1, 3, 6, 9, 11, 14, 22, 28 1 − 6, 8, 14, 16, 24 1 − 6, 8, 15, 18, 20 1 − 6, 10, 14, 22, 23 1 − 6, 10, 26, 30, 32 1 − 6, 11, 17, 28, 30 1 − 5, 7, 14, 17, 21, 25 1 − 5, 10, 17, 22, 25, 29 1 − 4, 6, 8, 21, 25, 30, 32 1 − 4, 6, 9, 12, 18, 27, 29 1 − 4, 6, 10, 17, 24, 26, 30 1 − 4, 6, 13, 17, 22, 23, 32 1 − 4, 6, 22, 24, 25, 30, 32 1 − 4, 7, 9, 13, 19, 22, 24 1 − 4, 7, 10, 11, 14, 21, 23 1 − 4, 7, 13, 15, 20, 22, 28 1 − 4, 7, 15, 19, 20, 27, 28 1 − 4, 8, 23, 24, 28, 30, 32 1 − 4, 9, 16, 17, 18, 21, 28 1 − 4, 9, 16, 18, 22, 26, 30 1 − 4, 10, 12, 14, 17, 18, 21 1 − 6, 10, 17 − 19, 22, 24 1 − 5, 7 − 9, 18, 29, 31, 32 1 − 5, 8, 10, 17, 20, 23, 30, 31 1 − 6, 16, 24, 25, 28, 30 1 − 5, 7, 20, 23, 25, 28, 30, 31 1 − 5, 11, 13, 14, 18, 26, 28, 32

7

C0 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C2 C2 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C1 C2 C2 C2

W W1 W1 W1 W1 W1 W1 W1 W1 W1 W1 W1 W1 W1 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2 W2

β 131 135 129 132 125 126 130 122 127 128 134 136 139 64 63 76 72 77 65 65 67 44 57 52 53 47 58 56 61 69 40 51 64 55 54 71 65 63 50 49 45 60 59 62 48 61 73 68

γ

1 1 1 1 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 1 0

3

Extremal Doubly-Even Codes of Length a Multiple of 24

Let C be a binary doubly-even self-dual [72, 36, 16] code. The existence of this code is an open question. In [2] Conway and Sloane have in their table that the highest minimum weight attainable for a length 70 singly-even code is 10 or 12, but this is an error as we shall see. This was also noticed later by Sloane in a private communication. A possible weight enumerator of a singly-even [70, 35, 14] code D has two parameters. It is a simple computation to find the weight enumerators of D and its shadow by Theorem 5 in [2]. This was first computed in [6], unfortunately the weight enumerator of the shadow was incorrectly reported. We give the corrected weight enumerators in Tables 5 and 6. Table 5: The Weight Distribution of a [70,35,14] Code Weight 0, 70 14, 56 16, 54 18, 52 20, 50 22, 48 24, 46 26, 44 28, 42 30, 40 32, 38 34, 36

Frequency 1 11730 150535 1345960 9393384 49991305 204312290 650311200 1627498400 3221810284 5066556495 6348487600

Table 6: The Weight Distribution of the Shadow of a [70,35,14] Code Weight 15, 55 19, 51 23, 47 27, 43 31, 39 35

Frequency 87584 7367360 208659360 2119532800 8314349120 13059745920

Note that any [70, 35, 14] code must have the same weight enumerator as the length 70 child of the extremal doubly-even [72, 36, 16] code. Thus the existence of an extremal doubly-even self-dual code of length 72 is equivalent to the existence of an extremal singlyeven self-dual code of length 70. 8

Lemma 3.1 For fixed k, the existence of a singly-even [24k − 2, 12k − 1, 4k + 2] code whose shadow has minimum weight 4k + 3 is equivalent to the existence of an extremal doubly-even code of length 24k. Proof. If any singly-even [24k − 2, 12k − 1, 4k + 2] code has a shadow with minimum weight 4k + 3, then since the weight enumerator of a [24k − 2, 12k − 1, 4k + 2] child of a doubly-even [24k, 12k, 4k + 4] is uniquely determined and the shadow of the child has minimum weight 4k + 3, the code is a child of an extremal doubly-even [24k, 12k, 4k + 4] code. Thus the existence of two codes is equivalent. 2 Remark. A similar argument can establish the corresponding results for extremal doublyeven codes of length 24k + 8. Now we examine the possibilities for weight enumerators of the singly-even [24k−2, 12k− 1, 4k + 2] codes, showing the details for the case when k = 4. Let us consider the possible weight enumerators of a singly-even [94, 47, 18] code C94 . By Theorem 5 in [2], the possible weight enumerators of C94 and its shadow S94 are determined. Since the minimum weight of C94 is 18, we have a0 = 1, a1 = −47, a2 = 846, a3 = −7379, a4 = 32806, a5 = −72850, a6 = 75764, a7 = −33511, a8 = 0 The weight enumerator WS of S94 is −

a10 11a11 7 a9 5a10 231a11 11 a11 3 y +( + )y + (− − − )y + · · · 524288 8192 262144 128 2048 524288

Let Bi be the coefficient of y i in WS . By Theorem 5 in [2], there exist three possibilities, namely (B3 , B7 ) = (1, 0), (0, 1) and (0, 0). It can be easily seen that the first two cases imply a negative or non-integral coefficient. We consider the last case that B3 = B7 = 0. Since it holds that a10 = a11 = 0, a9 11 9a9 15 WS = − y + y + ··· 128 64 a9 9 Since all the coefficients must be non-negative integers, − 128 ≥ 0 and 9a ≥ 0 then a9 = 0. 64 Thus the minimum weight of the shadow in the last case is 19. Similarly, we can get that the minimum weight of the shadow of a singly-even [24k − 2, 12k − 1, 4k + 2] code is 4k + 3 for 5 ≤ k ≤ 10. This computation is straightforward but tedious. By Lemma 3.1, we have the following: Theorem 3.2 The existence of an extremal doubly-even self-dual code of length 24k is equivalent to the existence of a singly-even self-dual [24k − 2, 12k − 1, 4k + 2] code for k ≤ 10.

9

Remark. It is well known that the existence of an extremal doubly-even self-dual code of length 24k and the existence of an extremal singly-even self-dual code of length 24k − 2 are equivalent when k = 1 and 2. The difficulty of the computation grows greatly at k = 10, we invite the reader to compute farther, to prove the following conjecture or find a counter-example. Conjecture 3.3 The existence of an extremal doubly-even self-dual code of length 24k is equivalent to the existence of a singly-even self-dual [24k − 2, 12k − 1, 4k + 2] code. Remark. All that is required to prove this conjecture is to show that a singly-even self-dual [24k − 2, 12k − 1, 4k + 2] code must have a shadow with minimum weight 4k + 3. Remark. For a singly-even [24k + 6, 12k + 3, 4k + 2] code, there are other possible weight enumerators than the shadow optimal codes when k = 1, 2 and 3. Hence there is nothing like the above conjecture happening. Namely the shadow optimal code of length 24k + 6 is not the only possible singly-even [24k + 6, 12k + 3, 4k + 2] code. Acknowledgment. The authors would like to thank Professors Eiichi Bannai and Vera Pless for their helpful comments on an earlier version of this correspondence, and the referee for useful comments.

References [1] S. Buyuklieva and I. Boukliev, “Extremal self-dual codes with an automorphism of order 2”, IEEE Trans. Inform. Theory, vol. IT-44, pp. 323–328, 1998. [2] J.H. Conway and N.J.A. Sloane, “A new upper bound on the minimal distance of self-dual codes,” IEEE Trans. Inform. Theory, vol. IT-36, pp. 1319–1333, 1990. [3] S.T. Dougherty and M. Harada, “Shadow optimal self-dual codes,” Kyushu J. Math., (to appear). [4] T.A. Gulliver and M. Harada, “ Classification of extremal double circulant self-dual codes of lengths 64 to 72,” Designs, Codes and Cryptogr., vol. 13, pp. 257–269, 1998. [5] M. Harada, “ The existence of a self-dual [70, 35, 12] code and formally self-dual codes,” Finite Fields and Their Appl., vol. 3, pp. 131–139, 1997. [6] G.T. Kennedy and V. Pless, “ A coding theoretic approach to extending designs,” Discrete Math., vol. 142, pp. 155–168, 1995. [7] N.J.A. Sloane, “ Is there a (72, 36) d = 16 self-dual code?” IEEE Trans. Inform. Theory, vol. IT-19, p. 251, 1973. 10