New Singly and Doubly Even Binary [72,36,12] Self-Dual Codes from M_2(R)G -- Group Matrix Rings
aa r X i v : . [ c s . I T ] F e b New Singly and Doubly Even Binary[72 , ,
12] Self-Dual Codes from M ( R ) G -Group Matrix Rings Adrian KorbanDepartment of Mathematical and Physical SciencesUniversity of ChesterThornton Science Park, Pool Ln, Chester CH2 4NU, EnglandSerap S¸ahinkayaTarsus University, Faculty of EngineeringDepartment of Natural and Mathematical SciencesMersin, TurkeyDeniz UstunTarsus University, Faculty of EngineeringDepartment of Computer EngineeringMersin, TurkeyFebruary 26, 2021
Abstract
In this work, we present a number of generator matrices of theform [ I n | τ k ( v )] , where I kn is the kn × kn identity matrix, v is anelement in the group matrix ring M ( R ) G and where R is a finitecommutative Frobenius ring and G is a finite group of order 18. Weemploy these generator matrices and search for binary [72 , ,
12] self-dual codes directly over the finite field F . As a result, we find 134Type I and 1 Type II codes of this length, with parameters in theirweight enumerators that were not known in the literature before. Wetabulate all of our findings. Introduction
A search for new binary self-dual codes of different lengths is still an ongoingresearch area in algebraic coding theory. Many researchers have employedvarious techniques to search for binary self-dual codes of different lengthswith new parameters in their weight enumerators. A classical technique isto consider a generator matrix of the form [ I n | A n ] , where I n is the n × n identity matrix and A n is some n × n matrix with entries from a finite field F . Of course, if we were to define the matrix A n in terms of n independentvariables, then only for the finite field F we would have a search field of 2 n which is not practical. Therefore, many researchers have considered matrices A n that are fully defined by the elements appearing in the first row - thisreduces the search field from 2 n to 2 n . For example, one can consider thematrix A n to be a circulant or a reverse circulant matrix.In [12], T. Hurley introduced a map σ ( v ) , where v is an element in thegroup ring RG with | G | = n, that sends v to an n × n matrix that is fullydefined by the elements appearing in the first row - these elements are fromthe ring R or finite field F q . By employing different groups G one can obtaindifferent n × n matrices as images under the map σ and this is the advantageof this map. In [9], the authors consider a generator matrix of the form[ I n | σ ( v )] for various groups G to search for new binary self-dual codes oflength 68 with a success. Recently in [4], the authors extended the map σ ( v )so that v ∈ RG gets sent to more complex n × n matrices that are also fullydefined by the elements appearing in the first row. They name this map asΩ( v ) and call the corresponding n × n matrices, the composite matrices -please see [4] for details. In [5], generator matrices of the form [ I n | Ω( v )] areconsidered to search for binary self-dual codes.The above two maps, σ and Ω , both send an element v from the group ring RG to an n × n matrices that are fully defined by the elements appearing inthe first rows. Recently in [8], the authors extended the map σ and consideredelements from the group matrix ring M k ( R ) G rather than elements from thegroup ring RG.
They defined a map that sends an element from the groupmatrix ring M k ( R ) G to a kn × kn matrix over the ring R. They called thismap τ k ( v ) - please see [8] for details. The advantage of this map is thatit does not only depend on the choice of the group G, but it also dependson the form of the elements from the matrix ring M k ( R ) , that is, the formof the k × k matrices over R. In this work, we employ the map τ k ( v ) andconsider generator matrices of the form [ I kn | τ k ( v )] with k = 2 and groups of2rder 18 to search for binary self-dual codes with parameters [72 , , . Wefind many such codes with weight enumerators that were not known in theliterature before.The rest of the work is organized as follows. In Section 2, we give prelim-inary definitions and results on self-dual codes, special matrices, group ringsand we also recall the the map τ k ( v ) that was defined in [8]. In Section 3,we present a number of generator matrices of the form [ I kn | τ k ( v )] for k = 2and groups of order 18. For each generator matrix, we fix the 2 × , , . As a result we find134 Type I and 1 Type II binary [72 , ,
12] self-dual codes with parametersin their weight enumerators that were not previously known. We tabulateour results, stating clearly the parameters of the obtained codes and theirorders of the automorphism group. We finish with concluding remarks anddirections for possible future research.
We begin by recalling the standard definitions from coding theory. A code C of length n over a Frobenius ring R is a subset of R n . If the code is asubmodule of R n then we say that the code is linear. Elements of the code C are called codewords of C . Let x = ( x , x , . . . , x n ) and y = ( y , y , . . . , y n )be two elements of R n . The duality is understood in terms of the Euclideaninner product, namely: h x , y i E = X x i y i . The dual C ⊥ of the code C is defined as C ⊥ = { x ∈ R n | h x , y i E = 0 for all y ∈ C } . We say that C is self-orthogonal if C ⊆ C ⊥ and is self-dual if C = C ⊥ . An upper bound on the minimum Hamming distance of a binary self-dualcode was given in [16]. Specifically, let d I ( n ) and d II ( n ) be the minimumdistance of a Type I (singly-even) and Type II (doubly-even) binary code of3ength n , respectively. Then d II ( n ) ≤ ⌊ n ⌋ + 4and d I ( n ) ≤ ( ⌊ n ⌋ + 4 if n
22 (mod 24)4 ⌊ n ⌋ + 6 if n ≡
22 (mod 24) . Self-dual codes meeting these bounds are called extremal . Throughoutthe text, we obtain extremal binary codes of different lengths. Self-dual codeswhich are the best possible for a given set of parameters is said to be optimal.Extremal codes are necessarily optimal but optimal codes are not necessarilyextremal.
To understand the form of the generator matrices which we define later inthis work, we recall some basic definitions of some special matrices and theoryon group rings.A circulant matrix is one where each row is shifted one element to theright relative to the preceding row. We label the circulant matrix as A = circ ( α , α . . . , α n ) , where α i are ring elements. The transpose of a matrix A, denoted by A T , is a matrix whose rows are the columns of A, i.e., A Tij = A ji . A symmetric matrix is a square matrix that is equal to its transpose. Apersymmetric matrix is a square matrix which is symmetric with respect tothe northeast-to-southwest diagonal. Later in this work, we only consider2 × persym ( a , a , a ) = (cid:18) a a a a (cid:19) . Let R be a ring, then if R has an identity 1 R , we say that u ∈ R is aunit in R if and only if there exists an element w ∈ R with uw = 1 R . Whilegroup rings can be given for infinite rings and infinite groups, we are onlyconcerned with group rings where both the ring and the group are finite. Let G be a finite group of order n , then the group ring RG consists of P ni =1 α i g i , α i ∈ R , g i ∈ G. n X i =1 α i g i + n X i =1 β i g i = n X i =1 ( α i + β i ) g i . (2.1)The product of two elements in a group ring is given by n X i =1 α i g i ! n X j =1 β j g j ! = X i,j α i β j g i g j . (2.2)It follows that the coefficient of g k in the product is P g i g j = g k α i β j . τ k ( v ) and generator matrices of the form [ I kn | τ k ( v )] We now recall the map τ k ( v ) , where v ∈ M k ( R ) G and where M k ( R ) is anon-commutative Frobenius matrix ring and G is a finite group of order n, that was introduced in [8].Let v = A g g + A g g + · · · + A g n g n ∈ M k ( R ) G, that is, each A g i is a k × k matrix with entries from the ring R. Define the block matrix σ k ( v ) ∈ ( M k ( R )) n to be σ k ( v ) = A g − g A g − g A g − g . . . A g − g n A g − g A g − g A g − g . . . A g − g n ... ... ... ... ... A g − n g A g − n g A g − n g . . . A g − n g n . (2.3)We note that the element v is an element of the group matrix ring M k ( R ) G. Construction 1
For a given element v ∈ M k ( R ) G, we define the follow-ing code over the matrix ring M k ( R ): C k ( v ) = h σ k ( v ) i . (2.4)Here the code is generated by taking the all left linear combinations of therows of the matrix with coefficients in M k ( R ) . Construction 2
For a given element v ∈ M k ( R ) G, we define the fol-lowing code over the ring R . Construct the matrix τ k ( v ) by viewing eachelement in a k by k matrix as an element in the larger matrix. B k ( v ) = h τ k ( v ) i . (2.5)5ere the code B k ( v ) is formed by taking all linear combinations of the rowsof the matrix with coefficients in R . In this case the ring over which the codeis defined is commutative so it is both a left linear and right linear code.We note that the map τ k ( v ) does not only depend on the choice of thegroup G and the ring R, but also on the structure of the k × k matrices. Laterin this work, we employ this map and consider generator matrices of the form[ I kn | τ k ( v )] for groups of order 18 and for k = 2 . That is, we consider 2 × I kn | τ k ( v )] from [8]. Lemma 2.1.
Let G be a group of order n and v = A g + A g + · · · + A n g n be an element of the group matrix ring M k ( R ) G. The matrix [ I kn | τ k ( v )] generates a self-dual code over R if and only if τ k ( v ) τ k ( v ) T = − I kn . Recall that the canonical involution ∗ : RG → RG on a group ring RG is given by v ∗ = P g a g g − , for v = P g a g g ∈ RG.
Also, recall that there isa connection between v ∗ and v when we take their images under the map σ, given by σ ( v ∗ ) = σ ( v ) T . (2.6)The above connection can be extended to the group matrix ring M k ( R ) G. Namely, let ∗ : M k ( R ) G → M k ( R ) G be the canonical involution on the groupmatrix ring M k ( R ) G given by v ∗ = P g A g g − , for v = P g A g g ∈ M k ( R ) G where A g are the k × k blocks. Then we have the following connection between v ∗ and v under the map τ k : τ k ( v ∗ ) = τ k ( v ) T . (2.7) Lemma 2.2.
Let R be a finite commutative ring. Let G be a group of order n with a fixed listing of its elements. Then the map τ k : v → M ( R ) kn is abijective ring homomorphism. Now, combining together Lemma 2.1, Lemma 2.2 and the fact that τ k ( v ) = − I kn if and only if v = − I k , we get the following corollary. Corollary 2.3.
Let M k ( R ) G be a group matrix ring, where M k ( R ) is a non-commutative Frobenius matrix ring. For v ∈ M k ( R ) G, the matrix [ I kn | τ k ( v )] generates a self-dual code over R if and only if vv ∗ = − I k . In particular v has to be a unit. When we restrict our attention to a matrix ring of characteristic 2, wehave that − I k = I k , which leads to the following further corollary:6 orollary 2.4. Let M k ( R ) G be a group matrix ring, where M k ( R ) is anon-commutative Frobenius matrix ring of characteristic 2. Then the ma-trix [ I kn | τ k ( v )] generates a self-dual code over R if and only if v satisfies vv ∗ = I k , namely v is a unitary unit in M k ( R ) G. In this section, we define generator matrices of the form [ I n | τ ( v )] where v ∈ M ( R ) G, for groups of order 18 and some 2 × G = h x, y | x = y = 1 , x y = x − i ∼ = D . Also, let v = P i =0 P j =0 A i +8 j y j x i ∈ M ( R ) D . Then: τ ( v ) = (cid:18) A BB A (cid:19) where A = CIRC ( A , A , A , . . . , A ) ,B = REV CIRC ( A , A , A , . . . , A ) , and where A i ∈ M ( R ) . Now we define five generator matrices of thefollowing forms:1. G = [ I | τ ( v )] , (3.1)with A = circ ( a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = circ ( a , a ) . G ′ = [ I | τ ( v )] , (3.2)with A = circ ( a , a ) , . . . , A = circ ( a , a ) ,A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) . G ′′ = [ I | τ ( v )] , (3.3)with A = circ ( a , a ) , A = persym ( a , a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = persym ( a , a , a ) , A = circ ( a , a ) ,A = persym ( a , a , a ) , A = circ ( a , a ) ,A = persym ( a , a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = persym ( a , a , a ) , G ′′′ = [ I | τ ( v )] , (3.4)with A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) ,A = circ ( a , a ) , . . . , A = circ ( a , a ) . G ′′′′ = [ I | τ ( v )] , (3.5)with A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) ,. . .A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) . II. Let G = h x, y | x = y = 1 , x y = x − i ∼ = D . Also, let v = P i =0 P j =0 A i +8 j x i y j ∈ M ( R ) D . Then: τ ( v ) = (cid:18) A BB T A T (cid:19) A = CIRC ( A , A , A , . . . , A ) ,B = CIRC ( A , A , A , . . . , A ) , and where A i ∈ M ( R ) . Now we define three generator matrices of thefollowing forms:1. G = [ I | τ ( v )] , (3.6)with A = circ ( a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = circ ( a , a ) . G ′ = [ I | τ ( v )] , (3.7)with A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) ,A = circ ( a , a ) , . . . , A = circ ( a , a ) . G ′′ = [ I | τ ( v )] , (3.8)with A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) ,. . .A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) . III. Let G = h x | x = 1 i C , . Also, let v = P i =0 P j =0 A i +3 j x i + j ∈ M ( R ) C . Then: τ ( v ) = A B C D E FF ′ A B C D EE ′ F ′ A B C DD ′ E ′ F ′ A B CC ′ D ′ E ′ F ′ A BB ′ C ′ D ′ E ′ F ′ A A = CIRC ( A , A , A ) , B = CIRC ( A , A , A ) ,B ′ = CIRC ( A , A , A ) , C = CIRC ( A , A , A ) ,C ′ = CIRC ( A , A , A ) , D = CIRC ( A , A , A ) ,D ′ = CIRC ( A , A , A ) , E = CIRC ( A , A , A ) ,E ′ = CIRC ( A , A , A ) , F = CIRC ( A , A , A ) ,F ′ = CIRC ( A , A , A ) , and where A i ∈ M ( R ) . Now we define a generator matrix of the fol-lowing form:1. G = [ I | τ ( v )] , (3.9)with A = circ ( a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = circ ( a , a ) . IV. Let G = h x, y | x = y = 1 , xy = yx i ∼ = C × C . Also, let v = P i =0 P j =0 A i +6 j x i y j ∈ M ( R )( C × C ) . Then: τ ( v ) = A B CC A BB C A where A = CIRC ( A , A , . . . , A ) , B = CIRC ( A , A , . . . , A ) ,C = CIRC ( A , A , . . . , A ) , and where A i ∈ M ( R ) . Now we define two generator matrices of thefollowing forms: 10. G = [ I | τ ( v )] , (3.10)with A = circ ( a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = circ ( a , a ) . G ′ = [ I | τ ( v )] , (3.11)with A = persym ( a , a , a ) , . . . , A = persym ( a , a , a ) ,A = circ ( a , a ) , . . . , A = circ ( a , a ) . V. Let G = h x, y | x = y = 1 , xy = yx i ∼ = C × C . Also, let v = P i =0 P j =0 A i + j x i y j ∈ M ( R )( C × C ) . Then: τ ( v ) = A B C D E FF A B C D EE F A B C DD E F A B CC D E F A BB C D E F A where A = CIRC ( A , A , A ) , B = CIRC ( A , A , A ) ,. . . ,E = CIRC ( A , A , A ) , F = CIRC ( A , A , A ) , and where A i ∈ M ( R ) . Now we define a generator matrix of the fol-lowing form:1. G = [ I | τ ( v )] , (3.12)with A = circ ( a , a ) , A = circ ( a , a ) ,. . . ,A = circ ( a , a ) , A = circ ( a , a ) .
11e note that in the above generator matrices, the choices for the 2 × × In this section, we employ the generator matrices defined in Section 3 andsearch for binary [72 , ,
12] self-dual codes.The possible weight enumerators for a Type I [72 , ,
12] codes are asfollows ([6]): W , = 1 + 2 βy + (8640 − γ ) y + (124281 − β + 384 γ ) y + . . .W , = 1 + 2 βy + (7616 − γ ) y + (134521 − β + 384 γ ) y + . . . where β and γ are parameters. The possible weight enumerators for Type II[72 , ,
12] codes are ([6]):1 + (4398 + α ) y + (197073 − α ) y + (18396972 + 66 α ) y + . . . where α is a parameter.Many codes for different values of α , β and γ have been constructed in[2, 3, 6, 7, 8, 10, 11, 13, 15, 17, 18, 19, 20]. For an up-to-date list of all knownType I and Type II binary self-dual codes with parameters [72 , ,
12] pleasesee [14].We now split the remaining of this section and tabulate our findingsaccording to the generator matrix we employ. We only list codes with pa-rameters in their weight enumerators that were not known in the literaturebefore. All the upcoming computational results were obtained by performingsearches using a particular algorithm technique (see [15] for details) in thesoftware package MAGMA ([1]).1. Generator matrices G , G ′ , G ′′ , G ′′′ and G ′′′′ In the generator matrix G , the matrix τ ( v ) is fully defined by thefirst row, for this reason, we only list the first row of the matrices A and B which we label as r A and r B respectively.12able 1: New Type I [72 , ,
12] Codes from G and R = F Type r A r B γ β | Aut ( C i ) | C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 0 93 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 0 111 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 0 132 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 0 138 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 0 144 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 0 150 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 0 174 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 0 198 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 0 309 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 0 345 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 0 366 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
1) 0 378 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 0 411 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 0 444 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 0 453 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 228 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 243 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 252 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 255 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 267 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 282 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 291 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 294 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 303 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 312 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
1) 18 318 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 321 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 18 330 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 18 333 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 339 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 348 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 351 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 360 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 363 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 366 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
1) 18 369 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 372 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 381 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 384 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 390 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 399 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 402 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 408 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 18 411 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 414 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 417 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 18 423 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 426 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 18 438 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 444 36 ype r A r B γ β | Aut ( C i ) | C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
1) 18 450 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 462 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 471 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 474 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 480 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 18 486 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 489 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 498 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 507 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
1) 18 516 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 18 525 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 18 540 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 36 393 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 36 399 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 402 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 36 444 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 36 453 72 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 36 462 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 36 477 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
1) 36 489 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 507 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 36 516 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 525 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 534 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 36 582 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 588 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 36 600 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 36 606 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 36 624 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 663 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 54 651 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 54 657 36
In the generator matrix G ′ , the matrix τ ( v ) is fully defined by the2 × A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A respectively. If the matrix A i is circulant, we only list the first row ofsuch matrix and if the matrix A i is persymmetric, we only list the threevariables that correspond to such matrix.Table 2: New Type I [72 , ,
12] Codes from G ′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 ,
0) (0 ,
1) (1 ,
0) (1 ,
0) (0 ,
0) (1 ,
1) (1 ,
0) (0 ,
0) (1 ,
1) 0 282 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (0 , ,
1) (0 , ,
0) (0 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
0) (0 , , ype r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
0) (0 ,
0) (1 ,
1) (0 ,
1) (0 ,
0) (1 ,
1) (0 ,
1) (0 ,
0) (0 ,
1) 9 192 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
1) (0 , ,
0) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
1) (1 ,
0) (0 ,
0) (0 ,
1) (1 ,
0) (1 ,
0) (0 ,
0) (0 ,
1) (1 ,
1) 9 210 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (1 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
1) (1 , ,
0) (0 , ,
1) (0 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
0) (1 ,
0) (1 ,
0) (1 ,
1) (1 ,
0) (1 ,
1) (0 ,
0) (1 ,
1) (1 ,
1) 9 225 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (0 , ,
0) (0 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
1) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
0) (1 ,
1) (1 ,
1) (0 ,
1) (1 ,
0) (1 ,
1) (0 ,
0) (0 ,
1) (0 ,
1) 9 228 18 r A r A r A r A r A r A r A r A r A (0 , ,
0) (0 , ,
1) (0 , ,
1) (1 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
1) (0 , ,
1) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
1) (0 ,
0) (1 ,
1) (0 ,
1) (0 ,
0) (1 ,
1) (0 ,
0) (0 ,
1) (0 ,
1) 9 255 18 r A r A r A r A r A r A r A r A r A (1 , ,
0) (1 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 ,
0) (1 ,
0) (0 ,
1) (0 ,
0) (0 ,
0) (0 ,
0) (1 ,
0) (1 ,
0) (0 ,
1) 9 258 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (0 , ,
1) (1 , ,
1) (0 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
0) (0 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 ,
0) (0 ,
1) (0 ,
0) (1 ,
0) (1 ,
0) (1 ,
0) (1 ,
1) (0 ,
1) (0 ,
1) 9 261 18 r A r A r A r A r A r A r A r A r A (0 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 ,
0) (1 ,
0) (0 ,
1) (0 ,
0) (1 ,
1) (0 ,
1) (0 ,
1) (1 ,
1) (1 ,
1) 9 270 18 r A r A r A r A r A r A r A r A r A (0 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
1) (0 , ,
0) (1 , ,
0) (0 , ,
0) (1 , ,
1) (1 , , ype r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
0) (0 ,
0) (1 ,
1) (1 ,
1) (1 ,
1) (1 ,
0) (1 ,
0) (1 ,
0) (0 ,
1) 9 282 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (0 , ,
0) (1 , ,
1) (1 , ,
0) (0 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 ,
1) (0 ,
0) (1 ,
1) (0 ,
0) (0 ,
1) (1 ,
1) (0 ,
0) (0 ,
1) (0 ,
1) 9 318 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
0) (0 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 ,
0) (0 ,
1) (1 ,
0) (0 ,
0) (1 ,
0) (0 ,
0) (1 ,
0) (1 ,
0) (1 ,
1) 9 336 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (1 , ,
1) (1 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
0) (0 , ,
1) (1 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 ,
1) (1 ,
0) (1 ,
0) (1 ,
0) (1 ,
1) (0 ,
0) (0 ,
1) (0 ,
1) (1 ,
0) 18 393 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (0 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
1) (0 , ,
0) (0 , ,
1) (1 , , In the generator matrix G ′′ , the matrix τ ( v ) is fully defined by the2 × A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A respectively. If the matrix A i is circulant, we only list the first row ofsuch matrix and if the matrix A i is persymmetric, we only list the threevariables that correspond to such matrix.Table 3: New Type I [72 , ,
12] Codes from G ′′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
1) (1 ,
1) (0 , ,
0) (1 ,
0) (0 , ,
0) (1 ,
0) (1 , ,
0) (0 ,
0) 9 252 18 r A r A r A r A r A r A r A r A r A (1 , ,
0) (0 ,
1) (0 , ,
1) (1 ,
0) (1 , ,
1) (1 ,
0) (1 , ,
1) (1 ,
0) (0 , , In the generator matrix G ′′′ , the matrix τ ( v ) is fully defined by the2 × A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A respectively. If the matrix A i is circulant, we only list the first row ofsuch matrix and if the matrix A i is persymmetric, we only list the threevariables that correspond to such matrix.16able 4: New Type I [72 , ,
12] Codes from G ′′′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
1) (0 , ,
0) 9 291 18 r A r A r A r A r A r A r A r A r A (0 ,
1) (0 ,
0) (1 ,
0) (0 ,
1) (0 ,
0) (1 ,
1) (0 ,
0) (0 ,
1) (0 , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
0) (1 , ,
1) (0 , ,
1) 9 300 18 r A r A r A r A r A r A r A r A r A (0 ,
1) (0 ,
1) (0 ,
0) (0 ,
0) (1 ,
1) (1 ,
1) (0 ,
1) (1 ,
1) (1 , In the generator matrix G ′′′′ , the matrix τ ( v ) is fully defined by the2 × A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A respectively. We only list thethree variables that correspond to such matrix.Table 5: New Type I [72 , ,
12] Codes from G ′′′′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
1) (1 , ,
0) (0 , ,
1) (1 , ,
1) (1 , ,
0) (0 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
0) 9 213 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
1) (0 , ,
1) (1 , ,
0) (0 , ,
1) (0 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
1) (0 , ,
1) (1 , ,
1) (0 , ,
0) (0 , ,
0) (1 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
1) 9 246 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
0) (1 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
1) (0 , ,
0) (1 , ,
0) (0 , ,
0) 9 279 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (1 , ,
0) (1 , ,
1) (0 , ,
0) (1 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
0) (0 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
1) 9 288 18 r A r A r A r A r A r A r A r A r A (1 , ,
0) (0 , ,
1) (1 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
0) (1 , ,
0) (0 , ,
1) (0 , , ype r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
1) (1 , ,
0) (0 , ,
0) (1 , ,
1) (1 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
1) (1 , ,
0) 9 315 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (1 , ,
1) (0 , ,
0) (0 , ,
1) (1 , ,
1) (0 , ,
0) (0 , ,
1) (1 , ,
1) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
1) (0 , ,
0) (0 , ,
0) (0 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
1) (1 , ,
1) 9 324 18 r A r A r A r A r A r A r A r A r A (0 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
1) (0 , ,
0) (0 , ,
0) (1 , ,
1) (1 , ,
1) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
1) 9 354 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (1 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
0) (1 , ,
0) (0 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
1) (0 , ,
0) 9 357 18 r A r A r A r A r A r A r A r A r A (0 , ,
0) (0 , ,
1) (1 , ,
1) (0 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
0) (0 , ,
1) (1 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
1) (0 , ,
0) (1 , ,
0) 27 354 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (0 , ,
1) (0 , ,
1) (0 , ,
1) (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
1) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
1) (0 , ,
1) (1 , ,
1) (0 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
0) 27 405 18 r A r A r A r A r A r A r A r A r A (1 , ,
0) (0 , ,
1) (0 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
0) (0 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
1) (0 , ,
1) (0 , ,
1) (0 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
0) (1 , ,
0) 27 444 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (0 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
0) (0 , ,
1) (1 , ,
0) (0 , ,
2. Generator matrices G , G ′ and G ′′ In the generator matrix G , the matrix τ ( v ) is fully defined by thefirst row, for this reason, we only list the first row of the matrices A and B which we label as r A and r B respectively.18able 6: New Type I [72 , ,
12] Codes from G and R = F Type r A r B γ β | Aut ( C i ) | C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 0 270 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 216 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 249 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 18 309 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 315 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 327 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 354 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 18 435 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 18 468 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 36 420 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
0) 36 567 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 36 615 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 54 642 36
In the generator matrix G ′ , the matrix τ ( v ) is fully defined by the2 × A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A respectively. If the matrix A i is circulant, we only list the first row ofsuch matrix and if the matrix A i is persymmetric, we only list the threevariables that correspond to such matrix.Table 7: New Type I [72 , ,
12] Codes from G ′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
1) (1 , ,
1) (0 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
0) 9 273 18 r A r A r A r A r A r A r A r A r A (0 ,
0) (1 ,
0) (0 ,
0) (0 ,
0) (1 ,
1) (1 ,
0) (0 ,
0) (1 ,
1) (0 , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
0) (1 , ,
0) (1 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
1) (0 , ,
0) 9 297 18 r A r A r A r A r A r A r A r A r A (1 ,
1) (1 ,
1) (0 ,
0) (0 ,
1) (0 ,
1) (0 ,
0) (0 ,
1) (0 ,
1) (0 , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
0) (0 , ,
1) (0 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
0) 9 306 18 r A r A r A r A r A r A r A r A r A (1 ,
0) (0 ,
0) (0 ,
1) (0 ,
0) (1 ,
1) (1 ,
1) (0 ,
1) (0 ,
0) (1 , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
1) (1 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
0) 9 309 18 r A r A r A r A r A r A r A r A r A (1 ,
0) (0 ,
0) (0 ,
0) (1 ,
0) (0 ,
1) (1 ,
0) (1 ,
0) (1 ,
1) (1 ,
19n the generator matrix G ′′ , the matrix τ ( v ) is fully defined by the 2 × A , A , A , . . . , A which we labelas r A , r A , r A , . . . , r A respectively. We only list the three variablesthat correspond to such matrix.Table 8: New Type I [72 , ,
12] Codes from G ′′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
1) (1 , ,
1) (1 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
0) 0 465 36 r A r A r A r A r A r A r A r A r A (1 , ,
0) (0 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
0) (1 , ,
1) (0 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
1) (0 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
0) 9 342 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (1 , ,
1) (1 , ,
0) (0 , ,
0) (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
0) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
0) (0 , ,
1) 9 345 18 r A r A r A r A r A r A r A r A r A (0 , ,
1) (0 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
1) (0 , ,
1) (1 , , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
1) (1 , ,
1) (0 , ,
1) (1 , ,
0) (0 , ,
0) (1 , ,
0) 9 375 18 r A r A r A r A r A r A r A r A r A (1 , ,
1) (1 , ,
0) (1 , ,
1) (0 , ,
1) (1 , ,
0) (1 , ,
0) (1 , ,
1) (0 , ,
1) (0 , ,
3. Generator matrix G In the generator matrix G , the matrix τ ( v ) is fully defined by thefirst row, for this reason, we only list the first row of the matrices A, B, C, D, E and F which we label as r A , r B , r C , r D , r E and r F respec-tively.Table 9: New Type I [72 , ,
12] Codes from G and R = F Type r A r B r C r D r E r F γ β | Aut ( C i ) | C W , (1 , , , , ,
1) (0 , , , , ,
1) (1 , , , , ,
0) (1 , , , , ,
0) (0 , , , , ,
0) (1 , , , , ,
1) 0 135 72
4. Generator matrix G In the generator matrix G , the matrix τ ( v ) is fully defined by thefirst row, for this reason, we only list the first row of the matrices A, B and C which we label as r A , r B and r C respectively.20able 10: New Type I [72 , ,
12] Codes from G and R = F Type r A r B r C γ β | Aut ( C i ) | C W , (1 , , , , , , , , , , ,
0) (0 , , , , , , , , , , ,
1) (1 , , , , , , , , , , ,
1) 0 165 72
In the generator matrix G ′ , the matrix τ ( v ) is fully defined by the2 × A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A respectively. If the matrix A i is circulant, we only list the first row ofsuch matrix and if the matrix A i is persymmetric, we only list the threevariables that correspond to such matrix.Table 11: New Type I [72 , ,
12] Codes from G ′ and R = F Type r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
1) (0 , ,
0) (0 , ,
0) (1 , ,
0) (1 , ,
0) (1 , ,
1) (1 , ,
0) (1 , ,
1) 18 297 36 r A r A r A r A r A r A r A r A r A (1 ,
0) (1 ,
0) (0 ,
0) (0 ,
0) (1 ,
0) (1 ,
1) (0 ,
1) (1 ,
1) (0 , r A r A r A r A r A r A r A r A r A γ β | Aut ( C i ) | C W , (1 , ,
0) (0 , ,
0) (0 , ,
1) (0 , ,
1) (0 , ,
1) (0 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
0) 36 378 36 r A r A r A r A r A r A r A r A r A (0 ,
1) (1 ,
1) (0 ,
0) (1 ,
0) (1 ,
0) (1 ,
1) (0 ,
1) (1 ,
0) (1 ,
5. Generator matrix G In the generator matrix G , the matrix τ ( v ) is fully defined by thefirst row, for this reason, we only list the first row of the matrices A, B, C, D, E and F which we label as r A , r B , r C , r D , r E and r E respec-tively.Table 12: New Type II [72 , ,
12] Codes from G and R = F r A r B r C r D r E r F α | Aut ( C i ) | C (0 , , , , ,
0) (0 , , , , ,
0) (1 , , , , ,
0) (0 , , , , ,
0) (1 , , , , ,
1) (0 , , , , , − In this paper, we defined generator matrices of the form [ I kn | τ k ( v )] - thisidea was first introduced in [8]. Such generator matrices depend on the21hoice of the group G and the form of the k × k matrices. We specificallyconsidered groups of order 18 and some 2 × k = 2 in ourgenerator matrices. We then employed our generator matrices to search forbinary [72 , ,
12] self-dual codes. We were able to construct Type I binary[72 , ,
12] self-dual codes with new weight enumerators in W , :( γ = 0 , β = { , , , , , , , , , , , , , , , , , , , } ) , ( γ = 9 , β = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , } ) , ( γ = 18 , β = { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } ) , ( γ = 27 , β = { , , } ) , ( γ = 36 , β = 378 , , , , , , , , , , , , , , , , , , , , , } )( γ = 54 , β = { , , } )and Type II binary [72 , ,
12] self-dual codes with new weight enumerators:( α = {− } ) , A suggestion for future work is to consider generator matrices of theform [ I kn | τ k ( v )] for groups of orders different than 18 and for values of k different than 2, to search for optimal binary self-dual codes of differentlengths. Another suggestion is to consider generator matrices of the form[ I kn | τ k ( v )] over different alphabets, for example, rings, and explore thebinary images of the codes under the Gray maps. References [1] W. Bosma, J. Cannon and C. Playoust, “The Magma algebra system. I.The user language”, J. Symbolic Comput., vol. 24, pp. 235–265, 1997.222] I. Bouyukliev, V. Fack and J. Winna, “Hadamard matrices of order36”, European Conference on Combinatorics, Graph Theory and Appli-cations, pp. 93–98, 2005.[3] R. Dontcheva, “New binary self-dual [70 , ,
12] and binary [72 , , n × n Matrices”, International Journal of Algebra and Computation, DOI:10.1142/S0218196721500223.[5] S.T. Dougherty, J. Gildea, A. Korban and A. Kaya, “Composite Matri-ces from Group Rings, Composite G -Codes and Constructions of Self-Dual Codes”, in submission .[6] S.T. Dougherty, T.A. Gulliver, M. Harada, “Extremal binary self dualcodes”, IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 2036–2047, 1997.[7] S.T. Dougherty, J-L. Kim and P. Sole, “Double circulant codes fromtwo class association schemes”, Advances in Mathematics of Communi-cations, vol. 1, no. 1, pp. 45–64, 2007.[8] S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, “Group MatrixRing Codes and Constructions of Self-Dual Codes”, arXiv:2102.00475.[9] J. Gildea, A. Kaya, R. Taylor and B. Yildiz, “Constructions for Self-dualCodes Induced from Group Rings”, Finite Fields Appl., vol. 51, (2018),71–92.[10] T.A. Gulliver, M. Harada, “On double circulant doubly-even self-dual[72 , ,
12] codes and their neighbors”, Austalas. J. Comb., vol. 40, pp.137-144, 2008.[11] M. Gurel, N. Yankov, “Self-dual codes with an automorphism of order17”, Mathematical Communications, vol. 21, no. 1, pp. 97–101, 2016.[12] T. Hurley, “Group Rings and Rings of Matrices”, Int. Jour. Pure andAppl. Math, vol. 31, no. 3, pp. 319–335, 2006.2313] A. Kaya, B. Yildiz and I. Siap, “New extremal binary self-dual codesof length 68 from quadratic residue codes over F + u F + u F ”, FiniteFIelds and Their Applications, vol. 29, pp. 160–177, 2014.[14] A. Korban, All known Type I and Type II[72 , ,
12] binary self-dual codes, available online at https://sites.google.com/view/adriankorban/binary-self-dual-codes .[15] A. Korban, S. Sahinkaya, D. Ustun, “A Novel Genetic Search SchemeBased on Nature – Inspired Evolutionary Algorithms for Self-DualCodes”, arXiv:2012.12248.[16] E.M. Rains, “Shadow Bounds for Self-Dual Codes”, IEEE Trans. Inf.Theory, vol. 44, pp. 134–139, 1998.[17] N. Tufekci, B. Yildiz, “On codes over R k,mk,m