New extremal singly even self-dual codes of lengths 64 and 66
aa r X i v : . [ m a t h . C O ] A ug New extremal singly even self-dual codes oflengths 64 and 66
Damyan Anev ∗ , Masaaki Harada † and Nikolay Yankov ‡ August 22, 2017
Abstract
For lengths 64 and 66, we construct extremal singly even self-dualcodes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new40 inequivalent extremal doubly even self-dual [64 , ,
12] codes withcovering radius 12 meeting the Delsarte bound.
A (binary) [ n, k ] code C is a k -dimensional vector subspace of F n , where F 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 calledan [ n, k, d ] code. The dual code C ⊥ of a code C of length n is defined as C ⊥ = { x ∈ F n | x · y = 0 for all y ∈ C } , where x · y is the standard innerproduct. A code C is called self-dual if C = C ⊥ . A self-dual code C is doubly ∗ Faculty of Mathematics and Informatics, Konstantin Preslavski University of Shumen,Shumen, 9712, Bulgaria. † Research Center for Pure and Applied Mathematics, Graduate School of InformationSciences, Tohoku University, Sendai 980–8579, Japan. ‡ Faculty of Mathematics and Informatics, Konstantin Preslavski University of Shumen,Shumen, 9712, Bulgaria. ven if all codewords of C have weight divisible by four, and singly even ifthere is at least one codeword x with wt( x ) ≡ n exists if and only if n is even, and a doubly evenself-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 C denote the subcode of C consisting of codewords x with wt( x ) ≡ shadow S of C is defined to be C ⊥ \ C . Shadows for self-dual codes were introduced byConway and Sloane [6] in order to give the largest possible minimum weightamong singly even self-dual codes, and to provide restrictions on the weightenumerators of singly even self-dual codes. The largest possible minimumweights among singly even self-dual codes of length n were given for n ≤ n ≤
72. Itis a fundamental problem to find which weight enumerators actually occurfor 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 ≤ ⌊ n ⌋ + 6 if n ≡
22 (mod 24), d ≤ ⌊ n ⌋ + 4 otherwise. Aself-dual code meeting the bound is called extremal .The aim of this note is to construct extremal singly even self-dual codeswith weight enumerators for which no extremal singly even self-dual codeswere previously known to exist. More precisely, we construct extremal singlyeven self-dual [64 , ,
12] codes with weight enumerators W , for β = 35, and W , for β ∈ { , , , , } (see Section 2 for W , and W , ). Thesecodes are constructed as self-dual neighbors of extremal four-circulant singlyeven self-dual codes. We construct extremal singly even self-dual [66 , , W , for β ∈ { , , , , } , and W , for β ∈ { , } (see Section 2 for W , and W , ). These codes are constructedfrom extremal singly even self-dual [64 , ,
12] codes by the method givenin [14]. We also demonstrate that there are at least 44 inequivalent extremaldoubly even self-dual [64 , ,
12] codes with covering radius 12 meeting theDelsarte bound.All computer calculations in this note were done with the help of thealgebra software
Magma [1] and the computer system Q-extensions [2].2
Weight enumerators of extremal singly evenself-dual codes of lengths 64 and 66
The possible weight enumerators W ,i and S ,i of extremal singly even self-dual [64 , ,
12] codes and their shadows are given in [6]: ( W , = 1 + (1312 + 16 β ) y + (22016 − β ) y + · · · ,S , = y + ( β − y + (3419 − β ) y + · · · , ( W , = 1 + (1312 + 16 β ) y + (23040 − β ) y + · · · ,S , = βy + (3328 − β ) y + · · · , where β are integers with 14 ≤ β ≤
104 for W , and 0 ≤ β ≤
277 for W , . Extremal singly even self-dual codes with weight enumerator W , are known for β ∈ (cid:26) , , , , , , , , , , , , , , , , , , , , , , (cid:27) (see [4], [10], [11] and [16]). Extremal singly even self-dual codes with weightenumerator W , are known for β ∈ (cid:26) , , . . . , , , , , , , , , , , , , , , , , , , , (cid:27) \ { , , , } (see [4], [10], [16] and [18]).The possible weight enumerators W ,i and S ,i of extremal singly evenself-dual [66 , ,
12] codes and their shadows are given in [7]: (cid:26) W , = 1 + (858 + 8 β ) y + (18678 − β ) y + · · · ,S , = βy + (10032 − β ) y + · · · , (cid:26) W , = 1 + 1690 y + 7990 y + · · · ,S , = y + 9680 y + · · · , (cid:26) W , = 1 + (858 + 8 β ) y + (18166 − β ) y + · · · ,S , = y + ( β − y + (10123 − β ) y + · · · , where β are integers with 0 ≤ β ≤
778 for W , and 14 ≤ β ≤
756 for W , . Extremal singly even self-dual codes with weight enumerator W , are known for β ∈ { , , . . . , , , , , } \ { , , , , } W , are known (see [8] and [15]). Extremal singly evenself-dual codes with weight enumerator W , are known for β ∈ { , , . . . , } \ { , , , , , } (see [9], [10], [11] and [12]). [64 , , codes An n × n circulant matrix has the following form: r r r · · · r n − r n − r r · · · r n − ... ... ... ... r r r · · · r , 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 [4 n, n ] code with generator matrixof the following form: (cid:18) I n A BB T A T (cid:19) , (1)where I n denotes the identity matrix of order n and A T denotes the transposeof A . It is easy to see that C is self-dual if AA T + BB T = I n . The codeswith generator matrices of the form (1) are called four-circulant .Two codes are equivalent if one can be obtained from the other by apermutation of coordinates. In this section, we give a classification of ex-tremal four-circulant singly even self-dual [64 , ,
12] codes. Our exhaus-tive search found all distinct extremal four-circulant singly even self-dual[64 , ,
12] codes, which must be checked further for equivalence to completethe classification. This was done by considering all pairs of 16 ×
16 circulantmatrices A and B satisfying the condition that AA T + BB T = I , the sumof the weights of the first rows of A and B is congruent to 1 (mod 4) and thesum of the weights is greater than or equal to 13. Since a cyclic shift of thefirst rows gives an equivalent code, we may assume without loss of generality4hat the last entry of the first row of B is 1. Then our computer searchshows that the above distinct extremal four-circulant singly even self-dual[64 , ,
12] codes are divided into 67 inequivalent codes.
Proposition 1.
Up to equivalence, there are extremal four-circulant singlyeven self-dual [64 , , codes. We denote the 67 codes by C ,i ( i = 1 , , . . . , C ,i ,the first rows r A (resp. r B ) of the circulant matrices A (resp. B ) in generatormatrices (1) are listed in Table 1. We verified that the codes C ,i have weightenumerator W , , where β are also listed in Table 1. [64 , , neighbors of C ,i Two self-dual codes C and C ′ of length n are said to be neighbors if dim( C ∩ C ′ ) = n/ −
1. Any self-dual code of length n can be reached from anyother by taking successive neighbors (see [6]). Since every self-dual code C of length n contains the all-one vector , C has 2 n/ − − D ofcodimension 1 containing . Since dim( D ⊥ /D ) = 2, there are two self-dualcodes rather than C lying between D ⊥ and D . If C is a singly even self-dualcode 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 , ,
12] codes byconsidering self-dual neighbors.For i = 1 , , . . . ,
67, we found all distinct extremal singly even self-dualneighbors of C ,i , which are equivalent to none of the 67 codes. Then weverified that these codes are divided into 385 inequivalent codes D ,i ( i =1 , , . . . , D ,i are constructed as h ( C ,j ∩ h x i ⊥ ) , x i . To save space, the values j , the supports supp( x ) of x , the values ( k, β ) inthe weight enumerators W ,k are listed in“ ”for the 385 codes. For extremal singly even self-dual [64 , ,
12] codes withweight enumerators for which no extremal singly even self-dual codes werepreviously known to exist, j , supp( x ) and ( k, β ) are list in Table 2. Hence,we have the following: Proposition 2.
There is an extremal singly even self-dual [64 , , codewith weight enumerator W , for β = 35 , and W , for β ∈ { , , , , } . , ,
12] codes
Codes r A r B βC , (0000001100111111) (0001011010101111) 0 C , (0000010101111101) (0010011010111011) 0 C , (0000011001101111) (0010110101011011) 0 C , (0000000001011111) (0001001100101011) 8 C , (0000000010101111) (0011011011110111) 8 C , (0000000011010111) (0000100110011011) 8 C , (0000000011010111) (0000101100010111) 8 C , (0000000011010111) (0011101110101111) 8 C , (0000000110111111) (0101101111111111) 8 C , (0000001001011101) (0001000101011011) 8 C , (0000001100011111) (0010101011011111) 8 C , (0000001100011111) (0010111011011011) 8 C , (0000001100111011) (0001101011101111) 8 C , (0000001101111111) (0011101111011111) 8 C , (0000010000111101) (0010111011011111) 8 C , (0000010001011111) (0001110101101111) 8 C , (0000010110111011) (0001101110001111) 8 C , (0000000100011111) (0010111111110011) 16 C , (0000000100111101) (0000101011000111) 16 C , (0000000110010111) (0001001111111111) 16 C , (0000000111001111) (0010101110111101) 16 C , (0000000111001111) (0010110110111011) 16 C , (0000001000101111) (0011101011110111) 16 C , (0000001011100011) (0010101111110111) 16 C , (0000001011100011) (0011011011111011) 16 C , (0000010010011111) (0010110011101111) 16 C , (0000011001101111) (0001001011011111) 16 C , (0000011011011111) (0010010101011101) 16 C , (0000011011100111) (0001011111001011) 16 C , (0000011101111111) (0101101110110111) 16 C , (0000101110111111) (0011101011110111) 16 C , (0000000000100111) (0001011101101011) 24 C , (0000000001011011) (0010010101101011) 24 C , (0000000100111111) (0001001000101011) 24 C , (0000000101001011) (0010010110011011) 24 C , (0000000101001011) (0010011001011011) 24 C , (0000000110111111) (0000001000100111) 24 C , (0000001001111111) (0010101111001011) 24 C , (0000001100011111) (0001010011111111) 24 C , (0000001100011111) (0001110011110111) 24 C , (0000010001011111) (0010101111001111) 24 C , (0000010001101111) (0011001110101111) 24 C , (0000010011101111) (0001011101100111) 24 C , (0000010101010111) (0001010111101111) 24 C , (0000010101010111) (0010110011111011) 24 C , (0000010101110111) (0000101111110011) 24 C , (0000010101110111) (0001011101101011) 24 C , (0000011011110111) (0101101110111111) 24 C , (0000000001001011) (0000111010110111) 32 C , (0000000001100111) (0001001111100011) 32 , ,
12] codes (con-tinued)
Codes r A r B βC , (0000001010111011) (0001011111100111) 32 C , (0000010101011111) (0001101111000111) 32 C , (0000010101111101) (0010110010110111) 32 C , (0000011010111111) (0000101110011101) 32 C , (0000101011101011) (0001011111001011) 32 C , (0000000000100111) (0001011010111011) 40 C , (0000000010101101) (0001001011011011) 40 C , (0000001000011101) (0000100101111011) 40 C , (0000001110011111) (0001010111101101) 40 C , (0000011000111111) (0001010111101101) 40 C , (0000011011001111) (0000101010111111) 40 C , (0000100111011111) (0001010101011011) 40 C , (0000001001101011) (0001010011001101) 48 C , (0000000001011011) (0001011000101111) 56 C , (0000010111011111) (0010100101011011) 56 C , (0000101110011101) (0001000101111111) 64 C , (0000000001011111) (0001011111110111) 72 Now we consider the extremal doubly even self-dual neighbors of C ,i ( i = 1 , , C ,i and C ,i are extremal doubly even self-dual[64 , ,
12] codes with covering radius 12 (see [4]). Thus, six extremal doublyeven self-dual [64 , ,
12] codes with covering radius 12 are constructed. Inaddition, among the 385 codes D ,i ( i = 1 , , . . . , D ,j have shadow of minimum weight 12, where j ∈ { , , , , , , , , , , , , , , , , , , } . The constructions of the 19 codes D ,j are listed in Table 2. Their twodoubly even self-dual neighbors D ,j and D ,j are extremal doubly evenself-dual [64 , ,
12] codes with covering radius 12. We verified that there arethe following equivalent codes among the four codes in [4], the six codes C ,i , C ,i and the 38 codes D ,j , D ,j , where D , ∼ = D , , D , ∼ = D , , D , ∼ = D , , D , ∼ = D , , where C ∼ = D means that C and D are equivalent, and there is no other pairof equivalent codes. Therefore, we have the following proposition. Proposition 3.
There are at least inequivalent extremal doubly even self-dual [64 , , codes with covering radius meeting the Delsarte bound. , ,
12] neighbors
Codes j supp( x ) ( k, β ) D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (1 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (2 , D , { , , , , , , , , , , , } (1 , D , { , , , , , , , , , , , , , , , } (2 ,
8n order to distinguish two doubly even neighbors D ,i and D ,i ( i =68 , , , x ) for the 8 codes, where D ,i and D ,i are constructed as h ( D ,i ∩ h x i ⊥ ) , x i .Table 3: Extremal doubly even self-dual [64 , ,
12] neighbors
Codes supp( x ) D , { , , , , , , , , , , , }D , { , , , , , , , , , , , }D , { , , , , , , , , , , , }D , { , , , , , , , , , , , }D , { , , , , , , , , , , , }D , { , , , , , , , , , , , }D , { , , , , , , , , , , , }D , { , , , , , , , , , , , } [64 , , codes and self-dual neighbors Using an approach similar to that given in Section 3, our exhaustive searchfound all distinct four-circulant singly even self-dual [64 , ,
10] codes. Thenour computer search shows that the distinct four-circulant singly even self-dual [64 , ,
10] codes are divided into 224 inequivalent codes.
Proposition 4.
Up to equivalence, there are four-circulant singly evenself-dual [64 , , codes. We denote the 224 codes by E ,i ( i = 1 , , . . . , r A (resp. r B ) of the circulant matrices A (resp. B ) in generatormatrices (1) can be obtained from“ ”.The following method for constructing self-dual neighbors was given in [4].For C = E ,i ( i = 1 , , . . . , M be a matrix whose rows are thecodewords of weight 10 in C . Suppose that there is a vector x of even weightsuch that M x T = T . (2)9hen C = h x i ⊥ ∩ C is a subcode of index 2 in C . We have self-dualneighbors h C , x i and h C , x + y i of C for some vector y ∈ C \ C , whichhave 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 , , . . . , C has two self-dual neighbors, where C is a doubly even[64 , ,
12] code. In this case, the two neighbors are automatically doublyeven. Hence, we have the following:
Proposition 5.
There is no extremal singly even self-dual [64 , , neigh-bor of E ,i for i = 1 , , . . . , . [66 , , codes The following method for constructing singly even self-dual codes was givenin [14]. Let C be a self-dual code of length n . Let x be a vector of oddweight. Let C denote the subcode of C consisting of all codewords whichare orthogonal to x . Then there are cosets C , C , C of C such that C ⊥ = C ∪ C ∪ C ∪ C , where C = C ∪ C and x + C = C ∪ C . It was shownin [14] that C ( x ) = (0 , , C ) ∪ (1 , , C ) ∪ (1 , , C ) ∪ (0 , , C ) (3)is a self-dual code of length n + 2. In this section, we construct new extremalsingly even self-dual codes of length 66 using this construction from theextremal singly even self-dual [64 , ,
12] codes obtained in Sections 3 and 4.Our exhaustive search shows that there are 1166 inequivalent extremalsingly even self-dual [66 , ,
12] codes constructed as the codes C ( x ) in (3)from the codes C ,i ( i = 1 , , . . . , W , for β ∈ { , , . . . , }\{ , } , 3 of them have weightenumerator W , for β ∈ { , , } , and 6 of them have weight enumerator W , . Extremal singly even self-dual [66 , ,
12] codes with weight enumer-ator W , for β ∈ { , , , } are constructed for the first time. For thefour weight enumerators W , as an example, codes C ,i with weight enumer-ators W are given ( i = 1 , , , β in W , thecodes C and the vectors x = ( x , x , . . . , x ) of C ( x ) in (3), where x j = 1( j = 33 , . . . , D ,i , we found more extremalsingly even self-dual [66 , ,
12] codes D ,j with weight enumerators for which10able 4: Extremal singly even self-dual [66 , ,
12] codes
Codes β W C ( x , . . . , x ) C , W , C , (01101101101010010111111010101100) C , W , C , (00001101100000011000110000011100) C , W , C , (00100110011011001001011100000010) C , W , C , (00001110110111110000011101000010) D , W , D , (10100011100100110111101010011111) D , W , D , (10111100111100000100101000100011) D , W , D , (10100101011110010011001101001101) no extremal singly even self-dual codes were previously known to exist. Forthe codes D ,j , we list in Table 4 the values β in the weight enumerators W , the codes C and the vectors x = ( x , x , . . . , x ) of C ( x ) in (3), where x i = 1 ( i = 33 , . . . , Proposition 6.
There is an extremal singly even self-dual [66 , , codewith weight enumerator W , for β ∈ { , , , , } , and weight enumer-ator W , for β ∈ { , } .Remark . The code D , has the smallest value β among known extremalsingly even self-dual [66 , ,
12] codes with weight enumerator W , . Acknowledgment.
This work was supported by JSPS KAKENHI GrantNumber 15H03633.
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