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Type I neighbors of extremal type II codes of length 40 derived from Hadamard matrices
Discrete Mathematics 259 (2002) 285 – 291
www.elsevier.com/locate/disc
Note
Type I neighbors of extremal type II codes of length 40 derived from Hadamard matrices Daniel B. Dalan1 Graduate School of Mathematics 33, Department of Mathematics, Kyushu University, Fukuoka 812-8581, Japan Received 20 June 2001; received in revised form 14 January 2002; accepted 28 January 2002
Abstract Self-dual codes C1 ; C2 of length n are called neighbors to each other if C1 ∩ C2 has dimension (n=2 − 1). With the aide of a computer, we search for Type I neighbors of some extremal Type II codes of length 40 which are derived from Hadamard matrices of order 20. As a result, we get extremal Type I neighbors (up to equivalence) that have weight enumerators c 2002 Elsevier Science B.V. All W40 (y) = 1 + (125 + 16)y8 + · · · with = 0; : : : ; 4; 6; 8; 10. rights reserved. Keywords: Type I codes; Type II codes; Neighbors; Hadamard matrices
1. Introduction Let F2n denote the linear space of all n-tuples over the 9nite 9eld F2 := GF(2). If C is a k-dimensional subspace of F2n , then C is called a binary [n; k] linear code. An element of C is called a codeword. The ordinary inner product of two vectors is de9ned by u · v :=
n
ui vi ;
i=1
where the vectors u := (u1 ; : : : ; un ); v := (v1 ; : : : ; vn ) are in F2n . 1
Graduate student of Kyushu University (Fukuoka, Japan) with scholarship grant from Asia—Japan Alumni (ASJA), International. E-mail address: [email protected] (D.B. Dalan). c 2002 Elsevier Science B.V. All rights reserved. 0012-365X/02/$ - see front matter PII: S 0 0 1 2 - 3 6 5 X ( 0 2 ) 0 0 3 7 3 - 4
286
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
The dual or orthogonal C ⊥ of C is the [n; n − k] linear code given by C ⊥ := {v ∈F2n | u · v =0 ∀u ∈C}: The Hamming weight of x (denoted by wt(x)) is the number of its nonzero coordinates. The smallest weight among nonzero codewords in C is called the minimum weight of C. If the minimum weight of C is known, then we refer to the code as an [n; k; d] code. Let C be a code. C is self-orthogonal (resp. self-dual) iK C is contained in its dual, i.e. C ⊆ C ⊥ (resp. iK C is equal to its dual, i.e. C = C ⊥ ). C is said to be self-dual singly even (resp., self-dual doubly even) if some codeword has weight ≡ 2 (mod 4) (resp., if all weights wt(x) ≡ 0 (mod 4); ∀x ∈C). Self-dual singly even (resp. self-dual doubly even) codes are referred to as Type I codes (resp. Type II codes). The set of monomial maps that send C to itself forms a group called the automorphism group of C (denoted by Aut(C)). For a Type II code of length n, it was shown in [12] that its minimum weight is at most d64[n=24] + 4. For a Type I code of length n, Rains [14] showed that the minimum weight d is upper bounded by d64[n=24] + 6 if n ≡ 22 (mod 24) and d64[n=24] + 4 otherwise. A code is called extremal if the minimum weight meets the bound with equality. In [5], a Type I code of length 40 with d= 8 has a weight enumerator given below W40 (y)=1 + (125 + 16)y8 + · · · : 2. Construction of the neighbors Self-dual codes C1 ; C2 of length n are called neighbors to each other if C1 ∩C2 has dimension (n=2−1). It is known that neighbors are a way of constructing new self-dual codes from known ones and such results can be found in [2,5]. Note that dim C1 = dim C2 =n=2 since C1 and C2 are self-dual. A self-dual code C of length n has 2n=2−1 − 1 hyperplanes (subcodes of codimension 1 in C) containing 1. For each such hyperplane H; dim H ⊥ =H = 2 holds. So there are three self-dual codes (including C) lying between H and H ⊥ . If, moreover C is Type II, then among the two other codes lying between H and H ⊥ , one is Type I and the other is Type II. Therefore, C has 2n=2−1 − 1 Type I neighbors and 2n=2−1 − 1 Type II neighbors. Some Type I codes arise in this way from a Type II code and its hyperplane. Indeed, if C1 is a Type I code, then there exists a unique doubly-even hyperplane H of C1 . The two neighbors C ; C of C1 such that C1 ∩C =C1 ∩C =H are both Type II. Next, we describe how the computer calculation was done on MAGMA. First, we compute the automorphism group of a Type II code C, then we construct its hyperplane, and 9nally, we construct its Type I neighbors. This procedure is adapted from Prof.
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291 Table 1 The Type I neighbors of C1 ; C2 , and C3 , and the number of codes with given minimum distance
Minimum distance
2
6
8
C1 C2 C3
3 3 3
29 82 133
6 14 7
287
Table 2 The Type I neighbors for C4 ; C5 ; C6 ; C7 , and C8 and the number of codes with given minimum distance
Minimum distance
2
6
8
C4
11
305
22
C5
16
563
19
C6
7
278
30
C7
12
527
35
C8
12
597
37
Akihiro Munemasa’s MAGMA computer program in [13] where all Type I codes of length 32 (there are 3210 of them) were completely enumerated. The problem on the equivalence of the extremal Type I codes constructed can be checked by MAGMA. The function is Boolean so when it returns a value true, the two codes being checked are equivalent, and thus we just choose one code to be counter checked against several other codes constructed. If the returned value is false, then the two codes are inequivalent and we only count the inequivalent codes for our enumeration. 3. The [40; 20; 8] Type II codes derived from Hadamard matrices A square matrix whose entries are ±1 that satis9es HH T = H T H = nI (I is the identity matrix and H T is the transpose of H ) is called a Hadamard matrix of order n. A Hadamard matrix having all +1 in its 9rst row and 9rst column is called a normalized Hadamard matrix. It is known that there are only three inequivalent Hadamard matrices of order 20. It is also known that it is possible to construct extremal Type II codes of length 40 using it is possible to construct extremal Type II codes of length 40 using Hadamard matrices of order 20 [16]. The following summarizes the computer results in searching for the Type I neighbors of Type II codes derived from normalized Hadamard matrices (see also Table 1). Proposition 1. Let C1 ; C2 ; C3 be the codes numbered 1, 22, and 71 in [3] constructed from the Hadamard matrices mentioned above. Then C1 has 38 Type I neighbors, C2 has 99 Type I neighbors, C3 has 143 Type I neighbors up to equivalence, respectively. Proposition 2. There are only 27 extremal Type I neighbors for extremal Type II codes of length 40 derived from normalized Hadamard matrices up to equivalence.
|Aut(C)|
C1; 1
768
2
C1; 2
16384
2
C1; 3
24576
6
C1; 4
1024
0
C1; 5
20480
0
C1; 6
1474560
10
C2; 1
256
2
C2; 2
48
2
C2; 3
512
4
C2; 4
48
0
C2; 5
48
0
C2; 6
160
0
C2; 7
128
0
C2; 8
512
0
C2; 9
256
0
Right half of the generator matrix of Ci; j (1777777), (3053711), (1777777), (3053711), (1777777), (3655133), (3613762), (3053711), (0015276), (2731210), (1606154), (3053711),
(3776023), (2607623), (3777000), (2607623), (3777000), (2607623), (1613015), (0763636), (1461170), (1273252), (1563517), (2607623),
(3741743), (2532325), (3140742), (2532325), (3146142), (2532325), (1624775), (0456330), (1456610), (2532325), (1554277), (2443506),
(3631635), (2515352), (3030634), (2515352), (3036034), (2515352), (3630616), (2515352), (1526766), (1161723), (1555522), (0701645),
(3607176), (2474471), (3606155), (2474471), (3606155), (2474471), (1762140), (2474471), (2164454), (2463546), (1563261), (2474471),
(3541037), (2463546), (3341015), (2463546), (3347615), (2463546), (3541037), (2463546), (1657147), (2362253), (3430614), (0706672),
(3231163), (2362253), (3431141), (2362253), (3437741), (2362253), (1355176), (2362253), (1127013), (2354507), (3231163), (2213470),
(3126532), (2354507), (3126532), (2354507), (3720310), (2354507), (1042527), (2354507), (2644033), (2313474), (1332025), (2225324),
(3125645), (2313474), (3125645), (2313474), (3723067), (2313474), (3125645), (2313474), (1233735), (2265334), (3054066), (0107163),
(3056266), (2265334) (3056266), (2265334) (3650444), (2265334) (3056266), (2265334) (1340316), (0762501) (1333152), (0071623)
(1777777), (3053711), (1777777), (3053711), (0352254), (1265051), (3425177), (3053711), (3653664), (3053711), (3612442), (1136424), (1777777), (3336427), (1777777), (3053711), (0023336), (1361062),
(3776023), (2613263), (3777000), (2613263), (1541740), (2532525), (3777000), (0541463), (3777000), (0737370), (3777000), (0776156), (3777000), (2576155), (3746163), (2613263), (1445773), (1275151),
(3741743), (2532525), (3650765), (2532525), (1576020), (2515552), (3740760), (0660325), (1664673), (0416436), (3740760), (0457610), (3740760), (2657613), (3771603), (2503446), (1472013), (1154617),
(3631635), (2515552), (3630616), (2515552), (1406156), (2466332), (1562016), (2515552), (3630616), (2515552), (1755523), (2515552), (3630616), (2515552), (3601775), (2524431), (1502165), (2515552),
(3607176), (2466332), (3606155), (2466332), (1430615), (2465645), (1554755), (2466332), (1722046), (2466332), (3606155), (2466332), (3563263), (2466332), (3637036), (2466332), (2152514), (2465645),
(3541037), (2465645), (3451032), (2465645), (2164514), (2362453), (1613637), (2465645), (1465124), (2465645), (3541037), (2465645), (3624301), (2465645), (3541037), (2465645), (1673744), (2362453),
(3225523), (2362453), (3335526), (2362453), (1013263), (2354307), (1177323), (2362453), (3225523), (2362453), (1340616), (2362453), (3140615), (2362453), (3225523), (2362453), (1117250), (2354307),
(3134271), (2354307), (3024274), (2354307), (1302531), (2307634), (3134271), (2354307), (3134271), (2354307), (3134271), (2354307), (3134271), (2354307), (3105312), (2354307), (2660630), (2307634),
(3123346), (2307634), (3123346), (2307634), (1315406), (2271174), (3123346), (2307634), (1007255), (2307634), (1046073), (2307634), (3246070), (2307634), (3112225), (2307634), (1211435), (2271174),
(3056466), (2271174) (3056466), (2271174) (1260326), (0425523) (1304266), (2271174) (1172575), (2271174) (1133753), (2271174) (3333750), (2271174) (3056466), (2271174) (2702027), (0754441)
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
Ci; j
288
Table 3 The orders of automorphism groups, values of W40 (y), and rows of the right half of the generator matrices of codes constructed in Proposition 2 in octal notation
12288
2
C2; 11
1536
4
C2; 12
1536
0
C2; 13
6144
6
C2; 14
640
0
C3; 1
24
2
C3; 2
12
3
C3; 3
8
0
C3; 4
12
0
C3; 5
6
0
C3; 6
20
0
C3; 7
36
1
(1777777), (3053711), (0346154), (2462132), (0370614), (1031037), (1777777), (3065051), (0421366), (1067753),
(3777000), (2613263), (1720511), (1275151), (1715726), (1276426), (3777000), (2613263), (1743042), (2613263),
(3740760), (2532525), (1717271), (2532525), (1722046), (1157360), (3740760), (2504265), (1774722), (1650176),
(3630616), (2513412), (1667307), (1173660), (1652130), (2515552), (3630616), (2523212), (1604654), (1677101),
(3606155), (2460272), (1651444), (2465645), (1664673), (2466332), (3606155), (2450472), (2550544), (2466332),
(3541037), (2465645), (2170614), (2362453), (2146154), (2362453), (3541037), (2453105), (1575075), (2465645),
(3225523), (2362453), (1272032), (2354307), (1247205), (2354307), (3225523), (2362453), (1211561), (2354307),
(3134271), (2354307), (2505452), (2307634), (1156557), (2307634), (3102531), (2354307), (2262660), (2307634),
(3125206), (2307634), (1174657), (2271174), (2524225), (2271174), (3115406), (2307634), (1117304), (2271174),
(3050526), (2271174) (1001177), (0431623) (2451505), (0407163) (3060326), (2271174) (2300077), (0356411)
(1777777), (3017750), (0634213), (2154234), (1777777), (3075517), (1777777), (3052264), (0556540), (2236567), (0567005), (1712504), (0206563), (1073062),
(3777000), (2623543), (1017027), (2616265), (3715247), (2641704), (3732534), (2666077), (2556237), (1360715), (1072254), (1336065), (1713732), (1336065),
(3746670), (2616265), (1020747), (2543256), (3722527), (2674022), (3740760), (2653751), (2561557), (1355033), (1045534), (1303743), (1724052), (2616265),
(3636706), (2543256), (1150631), (2515526), (3630616), (2543256), (3675322), (2543256), (1152677), (2515526), (1135442), (2543256), (1654124), (1056770),
(3522045), (2515526), (1244172), (2465631), (3524155), (2515526), (3524155), (2515526), (1246134), (2465631), (2734627), (2465631), (2055341), (2465631),
(3454023), (2465631), (1332114), (2354453), (3452133), (2465631), (3452133), (2465631), (1330152), (2354453), (2642641), (2354453), (1436601), (2354453),
(3303435), (2354453), (1463412), (2326332), (3361672), (2354453), (3346101), (2354453), (1461454), (2326332), (1406661), (2326332), (1367307), (2326332),
(3245307), (2326332), (1525320), (2271174), (3227140), (2326332), (3200633), (2326332), (1527366), (2271174), (1540153), (2271174), (1221435), (2271174),
(3131263), (2271174), (1651244), (2172705), (3153024), (2271174), (3131263), (2271174), (1653202), (2172705), (2321511), (2172705), (2440077), (2172705),
(3066466), (2172705) (1706441), (0143564) (3066466), (2172705) (3023152), (2172705) (1704407), (0221237) (1763632), (0210772) (2517672), (0571214)
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
C2; 10
289
290
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
Let C4 ; C5 ; C6 ; C7 and C8 be the extremal Type II codes of length 40 with code numbers 3, 4, 23, 5 and 24, respectively in [3]. These codes have automorphism groups of orders 214 · 3; 214 ; 212 · 3; 211 · 3; 211 · 3, respectively. Note that the codes chosen have the largest orders of automorphism groups among the codes constructed in [3]. With regards to C7 and C8 , they both have automorphism groups of order 211 · 3 but they come from diKerent designs. We chose them since we are also interested in knowing if there is any diKerence in the number of Type I neighbors of these codes. We list the summary of the Type I neighbors for codes C4 ; C5 ; C6 ; C7 , and C8 in Table 2. Proposition 3. Let C4 ; C5 ; C6 ; C7 , and C8 be extremal Type II codes with automorphism of order 214 3; 214 ; 212 3, and 211 3 from codes 3, 4, 23, 5 and 24 of [3], respectively. Then there are 338 Type I neighbors for C4 , 598 Type I neighbors for C5 , 315 Type I neighbors for C6 , 574 Type I neighbors for C7 , and 646 Type I neighbors for C8 up to equivalence, respectively. While checking for equivalence, we found out that two codes are equivalent (one Type I neighbor for C4 is equivalent to one Type I neighbor for C7 ) thus for the number of extremal Type I codes we have the following: Proposition 4. There are only 142 extremal Type I neighbors for extremal Type II codes C4 ; C5 ; C6 ; C7 , and C8 up to equivalence. Remark. According to Table X of [15], the number of inequivalent extremal Type I codes of length 40 is 22. However, note that there is an error since at least 39 inequivalent extremal Type I codes already exist in [4] (2 codes with |Aut(C)|= 7 and 37 codes with |Aut(C)|= 5). Now, if the extremal Type I codes of length 40 constructed in [5,6,7,8,10,11] are included in the list, then there are at least 50 inequivalent extremal Type I codes of length 40. Some new extremal Type I codes of length 40 have also been found recently in [1] and there are 22 of them. Thus, we can say that the present count of inequivalent extremal Type I codes of length 40 existing is 72. The extremal Type I codes of length 40 in [1,4,5,6,7,8,9,10,11] were inspected for equivalence to the codes constructed in this paper. It is good to know that none are equivalent and thus we have the following: Proposition 5. There are at least 72 + 219 = 291 inequivalent extremal Type I codes of length 40. We used MAGMA in computing for weight enumerators of the codes constructed. If we try to write down the order of automorphism groups and values of the weight enumerator for each of the 219 extremal Type I codes constructed, we will consume space so we will not write all of them down. However, we give the extremal codes constructed in Proposition 2. In Table 3, the 9rst column is the list of the extremal Type I neighbors labelled as Ci; j where i represents the Type II code and j its Type I
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
291
neighbor, the second column gives the order of automorphism groups, the third column gives the value of W40 (y), and the fourth column gives the rows of the right half of the generator matrices of each respective Ci; j in octal notation. Acknowledgements The author is very grateful to Prof. Akihiro Munemasa for the use of Prof. Munemasa’s MAGMA computer program in this work and his useful comment, and to Prof. Masaaki Harada for his comment on the improvement of this paper. References [1] S. Bouyuklieva, A method for constructing self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory 46 (2000) 496 – 504. [2] R.A. Brualdi, V. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory 37 (1991) 1222–1225. [3] F.C Bussemaker, V.D. Tonchev, New extremal doubly-even codes of length 40 derived from hadamard matrices of order 20, Discrete Math. 82 (1990) 317– 321. [4] S. Buyuklieva, V. Yorgov, Singly even self-dual codes of length 40, Des. Codes Cryptogr. 9 (1996) 131–141. [5] J.H. Conway, N.J.A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory 36 (1990) 1319 –1333. [6] S.T. Dougherty, T.A. Gulliver, M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory 43 (1997) 2036 –2047. [7] M. Harada, Existence of new extremal doubly-even and self-dual codes, Des. Codes Cryptogr. 8 (1996) 273 –283. [8] M. Harada, Weighing matrices and self-dual codes, Ars Combin. 47 (1997) 65 –73. [9] M. Harada, T.A. Gulliver, H. Kaneta, Classi9cation of extremal double circulant self-dual codes of length up to 62, Discrete Math. 188 (1998) 127–136. [10] M. Harada, H. Kimura, On extremal self-dual codes, Math. J. Okayama Univ. 37 (1995) 1–14. [11] M. Harada, V.D. Tonchev, Self-dual singly-even codes and hadamard matrices, in: Lecture Notes in Computer Science, Proceedings of AAECC 11, No. 948, New York, Springer, 1995, pp. 279 –284. [12] C.L. Mallows, N.J.A. Sloane, An upper bound for self-dual codes, Inform. and Control 22 (1973) 188 –200. [13] A. Munemasa, On the enumeration of self-dual codes, Summary Note of a Talk given at University of Tokyo, November 11, 2000. [14] E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory 44 (1998) 134 –139. [15] E. Rains, N.J.A Sloane, Self-dual codes, in: V.S. Pless, W.C. HuKman (Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, 1998. [16] V.D. Tonchev, Self-orthogonal designs and extremal doubly-even codes, J. Combin. Theory Ser. A 52 (1989) 197–205.
www.elsevier.com/locate/disc
Note
Type I neighbors of extremal type II codes of length 40 derived from Hadamard matrices Daniel B. Dalan1 Graduate School of Mathematics 33, Department of Mathematics, Kyushu University, Fukuoka 812-8581, Japan Received 20 June 2001; received in revised form 14 January 2002; accepted 28 January 2002
Abstract Self-dual codes C1 ; C2 of length n are called neighbors to each other if C1 ∩ C2 has dimension (n=2 − 1). With the aide of a computer, we search for Type I neighbors of some extremal Type II codes of length 40 which are derived from Hadamard matrices of order 20. As a result, we get extremal Type I neighbors (up to equivalence) that have weight enumerators c 2002 Elsevier Science B.V. All W40 (y) = 1 + (125 + 16)y8 + · · · with = 0; : : : ; 4; 6; 8; 10. rights reserved. Keywords: Type I codes; Type II codes; Neighbors; Hadamard matrices
1. Introduction Let F2n denote the linear space of all n-tuples over the 9nite 9eld F2 := GF(2). If C is a k-dimensional subspace of F2n , then C is called a binary [n; k] linear code. An element of C is called a codeword. The ordinary inner product of two vectors is de9ned by u · v :=
n
ui vi ;
i=1
where the vectors u := (u1 ; : : : ; un ); v := (v1 ; : : : ; vn ) are in F2n . 1
Graduate student of Kyushu University (Fukuoka, Japan) with scholarship grant from Asia—Japan Alumni (ASJA), International. E-mail address: [email protected] (D.B. Dalan). c 2002 Elsevier Science B.V. All rights reserved. 0012-365X/02/$ - see front matter PII: S 0 0 1 2 - 3 6 5 X ( 0 2 ) 0 0 3 7 3 - 4
286
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
The dual or orthogonal C ⊥ of C is the [n; n − k] linear code given by C ⊥ := {v ∈F2n | u · v =0 ∀u ∈C}: The Hamming weight of x (denoted by wt(x)) is the number of its nonzero coordinates. The smallest weight among nonzero codewords in C is called the minimum weight of C. If the minimum weight of C is known, then we refer to the code as an [n; k; d] code. Let C be a code. C is self-orthogonal (resp. self-dual) iK C is contained in its dual, i.e. C ⊆ C ⊥ (resp. iK C is equal to its dual, i.e. C = C ⊥ ). C is said to be self-dual singly even (resp., self-dual doubly even) if some codeword has weight ≡ 2 (mod 4) (resp., if all weights wt(x) ≡ 0 (mod 4); ∀x ∈C). Self-dual singly even (resp. self-dual doubly even) codes are referred to as Type I codes (resp. Type II codes). The set of monomial maps that send C to itself forms a group called the automorphism group of C (denoted by Aut(C)). For a Type II code of length n, it was shown in [12] that its minimum weight is at most d64[n=24] + 4. For a Type I code of length n, Rains [14] showed that the minimum weight d is upper bounded by d64[n=24] + 6 if n ≡ 22 (mod 24) and d64[n=24] + 4 otherwise. A code is called extremal if the minimum weight meets the bound with equality. In [5], a Type I code of length 40 with d= 8 has a weight enumerator given below W40 (y)=1 + (125 + 16)y8 + · · · : 2. Construction of the neighbors Self-dual codes C1 ; C2 of length n are called neighbors to each other if C1 ∩C2 has dimension (n=2−1). It is known that neighbors are a way of constructing new self-dual codes from known ones and such results can be found in [2,5]. Note that dim C1 = dim C2 =n=2 since C1 and C2 are self-dual. A self-dual code C of length n has 2n=2−1 − 1 hyperplanes (subcodes of codimension 1 in C) containing 1. For each such hyperplane H; dim H ⊥ =H = 2 holds. So there are three self-dual codes (including C) lying between H and H ⊥ . If, moreover C is Type II, then among the two other codes lying between H and H ⊥ , one is Type I and the other is Type II. Therefore, C has 2n=2−1 − 1 Type I neighbors and 2n=2−1 − 1 Type II neighbors. Some Type I codes arise in this way from a Type II code and its hyperplane. Indeed, if C1 is a Type I code, then there exists a unique doubly-even hyperplane H of C1 . The two neighbors C ; C of C1 such that C1 ∩C =C1 ∩C =H are both Type II. Next, we describe how the computer calculation was done on MAGMA. First, we compute the automorphism group of a Type II code C, then we construct its hyperplane, and 9nally, we construct its Type I neighbors. This procedure is adapted from Prof.
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291 Table 1 The Type I neighbors of C1 ; C2 , and C3 , and the number of codes with given minimum distance
Minimum distance
2
6
8
C1 C2 C3
3 3 3
29 82 133
6 14 7
287
Table 2 The Type I neighbors for C4 ; C5 ; C6 ; C7 , and C8 and the number of codes with given minimum distance
Minimum distance
2
6
8
C4
11
305
22
C5
16
563
19
C6
7
278
30
C7
12
527
35
C8
12
597
37
Akihiro Munemasa’s MAGMA computer program in [13] where all Type I codes of length 32 (there are 3210 of them) were completely enumerated. The problem on the equivalence of the extremal Type I codes constructed can be checked by MAGMA. The function is Boolean so when it returns a value true, the two codes being checked are equivalent, and thus we just choose one code to be counter checked against several other codes constructed. If the returned value is false, then the two codes are inequivalent and we only count the inequivalent codes for our enumeration. 3. The [40; 20; 8] Type II codes derived from Hadamard matrices A square matrix whose entries are ±1 that satis9es HH T = H T H = nI (I is the identity matrix and H T is the transpose of H ) is called a Hadamard matrix of order n. A Hadamard matrix having all +1 in its 9rst row and 9rst column is called a normalized Hadamard matrix. It is known that there are only three inequivalent Hadamard matrices of order 20. It is also known that it is possible to construct extremal Type II codes of length 40 using it is possible to construct extremal Type II codes of length 40 using Hadamard matrices of order 20 [16]. The following summarizes the computer results in searching for the Type I neighbors of Type II codes derived from normalized Hadamard matrices (see also Table 1). Proposition 1. Let C1 ; C2 ; C3 be the codes numbered 1, 22, and 71 in [3] constructed from the Hadamard matrices mentioned above. Then C1 has 38 Type I neighbors, C2 has 99 Type I neighbors, C3 has 143 Type I neighbors up to equivalence, respectively. Proposition 2. There are only 27 extremal Type I neighbors for extremal Type II codes of length 40 derived from normalized Hadamard matrices up to equivalence.
|Aut(C)|
C1; 1
768
2
C1; 2
16384
2
C1; 3
24576
6
C1; 4
1024
0
C1; 5
20480
0
C1; 6
1474560
10
C2; 1
256
2
C2; 2
48
2
C2; 3
512
4
C2; 4
48
0
C2; 5
48
0
C2; 6
160
0
C2; 7
128
0
C2; 8
512
0
C2; 9
256
0
Right half of the generator matrix of Ci; j (1777777), (3053711), (1777777), (3053711), (1777777), (3655133), (3613762), (3053711), (0015276), (2731210), (1606154), (3053711),
(3776023), (2607623), (3777000), (2607623), (3777000), (2607623), (1613015), (0763636), (1461170), (1273252), (1563517), (2607623),
(3741743), (2532325), (3140742), (2532325), (3146142), (2532325), (1624775), (0456330), (1456610), (2532325), (1554277), (2443506),
(3631635), (2515352), (3030634), (2515352), (3036034), (2515352), (3630616), (2515352), (1526766), (1161723), (1555522), (0701645),
(3607176), (2474471), (3606155), (2474471), (3606155), (2474471), (1762140), (2474471), (2164454), (2463546), (1563261), (2474471),
(3541037), (2463546), (3341015), (2463546), (3347615), (2463546), (3541037), (2463546), (1657147), (2362253), (3430614), (0706672),
(3231163), (2362253), (3431141), (2362253), (3437741), (2362253), (1355176), (2362253), (1127013), (2354507), (3231163), (2213470),
(3126532), (2354507), (3126532), (2354507), (3720310), (2354507), (1042527), (2354507), (2644033), (2313474), (1332025), (2225324),
(3125645), (2313474), (3125645), (2313474), (3723067), (2313474), (3125645), (2313474), (1233735), (2265334), (3054066), (0107163),
(3056266), (2265334) (3056266), (2265334) (3650444), (2265334) (3056266), (2265334) (1340316), (0762501) (1333152), (0071623)
(1777777), (3053711), (1777777), (3053711), (0352254), (1265051), (3425177), (3053711), (3653664), (3053711), (3612442), (1136424), (1777777), (3336427), (1777777), (3053711), (0023336), (1361062),
(3776023), (2613263), (3777000), (2613263), (1541740), (2532525), (3777000), (0541463), (3777000), (0737370), (3777000), (0776156), (3777000), (2576155), (3746163), (2613263), (1445773), (1275151),
(3741743), (2532525), (3650765), (2532525), (1576020), (2515552), (3740760), (0660325), (1664673), (0416436), (3740760), (0457610), (3740760), (2657613), (3771603), (2503446), (1472013), (1154617),
(3631635), (2515552), (3630616), (2515552), (1406156), (2466332), (1562016), (2515552), (3630616), (2515552), (1755523), (2515552), (3630616), (2515552), (3601775), (2524431), (1502165), (2515552),
(3607176), (2466332), (3606155), (2466332), (1430615), (2465645), (1554755), (2466332), (1722046), (2466332), (3606155), (2466332), (3563263), (2466332), (3637036), (2466332), (2152514), (2465645),
(3541037), (2465645), (3451032), (2465645), (2164514), (2362453), (1613637), (2465645), (1465124), (2465645), (3541037), (2465645), (3624301), (2465645), (3541037), (2465645), (1673744), (2362453),
(3225523), (2362453), (3335526), (2362453), (1013263), (2354307), (1177323), (2362453), (3225523), (2362453), (1340616), (2362453), (3140615), (2362453), (3225523), (2362453), (1117250), (2354307),
(3134271), (2354307), (3024274), (2354307), (1302531), (2307634), (3134271), (2354307), (3134271), (2354307), (3134271), (2354307), (3134271), (2354307), (3105312), (2354307), (2660630), (2307634),
(3123346), (2307634), (3123346), (2307634), (1315406), (2271174), (3123346), (2307634), (1007255), (2307634), (1046073), (2307634), (3246070), (2307634), (3112225), (2307634), (1211435), (2271174),
(3056466), (2271174) (3056466), (2271174) (1260326), (0425523) (1304266), (2271174) (1172575), (2271174) (1133753), (2271174) (3333750), (2271174) (3056466), (2271174) (2702027), (0754441)
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
Ci; j
288
Table 3 The orders of automorphism groups, values of W40 (y), and rows of the right half of the generator matrices of codes constructed in Proposition 2 in octal notation
12288
2
C2; 11
1536
4
C2; 12
1536
0
C2; 13
6144
6
C2; 14
640
0
C3; 1
24
2
C3; 2
12
3
C3; 3
8
0
C3; 4
12
0
C3; 5
6
0
C3; 6
20
0
C3; 7
36
1
(1777777), (3053711), (0346154), (2462132), (0370614), (1031037), (1777777), (3065051), (0421366), (1067753),
(3777000), (2613263), (1720511), (1275151), (1715726), (1276426), (3777000), (2613263), (1743042), (2613263),
(3740760), (2532525), (1717271), (2532525), (1722046), (1157360), (3740760), (2504265), (1774722), (1650176),
(3630616), (2513412), (1667307), (1173660), (1652130), (2515552), (3630616), (2523212), (1604654), (1677101),
(3606155), (2460272), (1651444), (2465645), (1664673), (2466332), (3606155), (2450472), (2550544), (2466332),
(3541037), (2465645), (2170614), (2362453), (2146154), (2362453), (3541037), (2453105), (1575075), (2465645),
(3225523), (2362453), (1272032), (2354307), (1247205), (2354307), (3225523), (2362453), (1211561), (2354307),
(3134271), (2354307), (2505452), (2307634), (1156557), (2307634), (3102531), (2354307), (2262660), (2307634),
(3125206), (2307634), (1174657), (2271174), (2524225), (2271174), (3115406), (2307634), (1117304), (2271174),
(3050526), (2271174) (1001177), (0431623) (2451505), (0407163) (3060326), (2271174) (2300077), (0356411)
(1777777), (3017750), (0634213), (2154234), (1777777), (3075517), (1777777), (3052264), (0556540), (2236567), (0567005), (1712504), (0206563), (1073062),
(3777000), (2623543), (1017027), (2616265), (3715247), (2641704), (3732534), (2666077), (2556237), (1360715), (1072254), (1336065), (1713732), (1336065),
(3746670), (2616265), (1020747), (2543256), (3722527), (2674022), (3740760), (2653751), (2561557), (1355033), (1045534), (1303743), (1724052), (2616265),
(3636706), (2543256), (1150631), (2515526), (3630616), (2543256), (3675322), (2543256), (1152677), (2515526), (1135442), (2543256), (1654124), (1056770),
(3522045), (2515526), (1244172), (2465631), (3524155), (2515526), (3524155), (2515526), (1246134), (2465631), (2734627), (2465631), (2055341), (2465631),
(3454023), (2465631), (1332114), (2354453), (3452133), (2465631), (3452133), (2465631), (1330152), (2354453), (2642641), (2354453), (1436601), (2354453),
(3303435), (2354453), (1463412), (2326332), (3361672), (2354453), (3346101), (2354453), (1461454), (2326332), (1406661), (2326332), (1367307), (2326332),
(3245307), (2326332), (1525320), (2271174), (3227140), (2326332), (3200633), (2326332), (1527366), (2271174), (1540153), (2271174), (1221435), (2271174),
(3131263), (2271174), (1651244), (2172705), (3153024), (2271174), (3131263), (2271174), (1653202), (2172705), (2321511), (2172705), (2440077), (2172705),
(3066466), (2172705) (1706441), (0143564) (3066466), (2172705) (3023152), (2172705) (1704407), (0221237) (1763632), (0210772) (2517672), (0571214)
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
C2; 10
289
290
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
Let C4 ; C5 ; C6 ; C7 and C8 be the extremal Type II codes of length 40 with code numbers 3, 4, 23, 5 and 24, respectively in [3]. These codes have automorphism groups of orders 214 · 3; 214 ; 212 · 3; 211 · 3; 211 · 3, respectively. Note that the codes chosen have the largest orders of automorphism groups among the codes constructed in [3]. With regards to C7 and C8 , they both have automorphism groups of order 211 · 3 but they come from diKerent designs. We chose them since we are also interested in knowing if there is any diKerence in the number of Type I neighbors of these codes. We list the summary of the Type I neighbors for codes C4 ; C5 ; C6 ; C7 , and C8 in Table 2. Proposition 3. Let C4 ; C5 ; C6 ; C7 , and C8 be extremal Type II codes with automorphism of order 214 3; 214 ; 212 3, and 211 3 from codes 3, 4, 23, 5 and 24 of [3], respectively. Then there are 338 Type I neighbors for C4 , 598 Type I neighbors for C5 , 315 Type I neighbors for C6 , 574 Type I neighbors for C7 , and 646 Type I neighbors for C8 up to equivalence, respectively. While checking for equivalence, we found out that two codes are equivalent (one Type I neighbor for C4 is equivalent to one Type I neighbor for C7 ) thus for the number of extremal Type I codes we have the following: Proposition 4. There are only 142 extremal Type I neighbors for extremal Type II codes C4 ; C5 ; C6 ; C7 , and C8 up to equivalence. Remark. According to Table X of [15], the number of inequivalent extremal Type I codes of length 40 is 22. However, note that there is an error since at least 39 inequivalent extremal Type I codes already exist in [4] (2 codes with |Aut(C)|= 7 and 37 codes with |Aut(C)|= 5). Now, if the extremal Type I codes of length 40 constructed in [5,6,7,8,10,11] are included in the list, then there are at least 50 inequivalent extremal Type I codes of length 40. Some new extremal Type I codes of length 40 have also been found recently in [1] and there are 22 of them. Thus, we can say that the present count of inequivalent extremal Type I codes of length 40 existing is 72. The extremal Type I codes of length 40 in [1,4,5,6,7,8,9,10,11] were inspected for equivalence to the codes constructed in this paper. It is good to know that none are equivalent and thus we have the following: Proposition 5. There are at least 72 + 219 = 291 inequivalent extremal Type I codes of length 40. We used MAGMA in computing for weight enumerators of the codes constructed. If we try to write down the order of automorphism groups and values of the weight enumerator for each of the 219 extremal Type I codes constructed, we will consume space so we will not write all of them down. However, we give the extremal codes constructed in Proposition 2. In Table 3, the 9rst column is the list of the extremal Type I neighbors labelled as Ci; j where i represents the Type II code and j its Type I
D.B. Dalan / Discrete Mathematics 259 (2002) 285 – 291
291
neighbor, the second column gives the order of automorphism groups, the third column gives the value of W40 (y), and the fourth column gives the rows of the right half of the generator matrices of each respective Ci; j in octal notation. Acknowledgements The author is very grateful to Prof. Akihiro Munemasa for the use of Prof. Munemasa’s MAGMA computer program in this work and his useful comment, and to Prof. Masaaki Harada for his comment on the improvement of this paper. References [1] S. Bouyuklieva, A method for constructing self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory 46 (2000) 496 – 504. [2] R.A. Brualdi, V. Pless, Weight enumerators of self-dual codes, IEEE Trans. Inform. Theory 37 (1991) 1222–1225. [3] F.C Bussemaker, V.D. Tonchev, New extremal doubly-even codes of length 40 derived from hadamard matrices of order 20, Discrete Math. 82 (1990) 317– 321. [4] S. Buyuklieva, V. Yorgov, Singly even self-dual codes of length 40, Des. Codes Cryptogr. 9 (1996) 131–141. [5] J.H. Conway, N.J.A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory 36 (1990) 1319 –1333. [6] S.T. Dougherty, T.A. Gulliver, M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory 43 (1997) 2036 –2047. [7] M. Harada, Existence of new extremal doubly-even and self-dual codes, Des. Codes Cryptogr. 8 (1996) 273 –283. [8] M. Harada, Weighing matrices and self-dual codes, Ars Combin. 47 (1997) 65 –73. [9] M. Harada, T.A. Gulliver, H. Kaneta, Classi9cation of extremal double circulant self-dual codes of length up to 62, Discrete Math. 188 (1998) 127–136. [10] M. Harada, H. Kimura, On extremal self-dual codes, Math. J. Okayama Univ. 37 (1995) 1–14. [11] M. Harada, V.D. Tonchev, Self-dual singly-even codes and hadamard matrices, in: Lecture Notes in Computer Science, Proceedings of AAECC 11, No. 948, New York, Springer, 1995, pp. 279 –284. [12] C.L. Mallows, N.J.A. Sloane, An upper bound for self-dual codes, Inform. and Control 22 (1973) 188 –200. [13] A. Munemasa, On the enumeration of self-dual codes, Summary Note of a Talk given at University of Tokyo, November 11, 2000. [14] E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory 44 (1998) 134 –139. [15] E. Rains, N.J.A Sloane, Self-dual codes, in: V.S. Pless, W.C. HuKman (Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, 1998. [16] V.D. Tonchev, Self-orthogonal designs and extremal doubly-even codes, J. Combin. Theory Ser. A 52 (1989) 197–205.