Group Matrix Ring Codes and Constructions of Self-Dual Codes
Steven Dougherty, Adrian Korban, Serap Sahinkaya, Deniz Ustun
aa r X i v : . [ c s . I T ] J a n Group Matrix Ring Codes and Constructionsof Self-Dual Codes
S. T. DoughertyUniversity of ScrantonScranton, PA, 18518, USAAdrian 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 2, 2021
Abstract
In this work, we study codes generated by elements that come fromgroup matrix rings. We present a matrix construction which we use togenerate codes in two different ambient spaces: the matrix ring M k ( R )and the ring R, where R is the commutative Frobenius ring. We showthat codes over the ring M k ( R ) are one sided ideals in the group matrixring M k ( R ) G and the corresponding codes over the ring R are G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. e employ this generator matrix to search for binary self-dual codeswith parameters [72 , ,
12] and find new singly-even and doubly-evencodes of this type. In particular, we construct 16 new Type I and 4new Type II binary [72 , ,
12] self-dual codes.
Self-dual codes are one of the most widely studied and interesting class ofcodes. They have been shown to have strong connections to unimodularlattices, invariant theory, and designs. In particular, binary self-dual codeshave been extensively studied and numerous construction techniques of self-dual codes have been used in an attempt to find optimal self-dual codes.In this work, we give a new construction of self-dual codes motivated bythe constructions given in [6] and [12]. In our construction, we use grouprings where the ring is a ring of matrices to construct generator matrices ofself-dual codes. The main point of this construction is to find codes thatother techniques have missed. We construct numerous new self-dual codesusing this technique.We begin with some definitions. A code C over an alphabet A of length n is a subset of A n . We say that the code is linear over A if A is a ringand C is a submodule. This implies that when A is a finite field, then C is a vector space. We attach to the ambient space the standard Euclideaninner-product, that is [ v , w ] = P v i w i . When A is commutative, we definethe orthogonal to this inner-product as C ⊥ = { w | [ w , v ] = 0 , ∀ v ∈ C } . Ifthe ring A is not commutative, then we say that the code is either left linearor right linear depending if it is a left or right module. In this scenario,we have two orthogonals, namely L ( C ) = { w | [ w , v ] = 0 , ∀ v ∈ C } and R ( C ) = { w | [ v , w ] = 0 , ∀ v ∈ C } . If the ring is not commutative then thesetwo codes are not necessarily equal, and in general will not be. Moreover, L ( C ) is a left linear code and R ( C ) is a right linear code. If the ring iscommutative, then L ( C ) = R ( C ) = C ⊥ . It is known that if C is a left linearcode over a Frobenius ring R then | C ||R ( C ) | = | A n | and if C is a rightlinear code over a Frobenius ring R then | C ||L ( C ) | = | A n | . For commutativerings, this gives that | C || C ⊥ | = | A n | or dim( C ) + dim( C ⊥ ) = n as usual.For a complete description of codes over commutative rings see [4]. For adescription of codes over non-commutative rings see [5]. Throughout thiswork we assume that every ring has a multiplicative identity and is finite.2n 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 oflength 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 .In this work, we shall use the theory of group rings to build codes. Weshall give the necessary definitions for this study. Let R be a ring, then if R has an identity 1 R , we say that u ∈ R is a unit in R if and only if thereexists an element w ∈ R with uw = 1 R . 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. Addition in the group ring is done by coordinate addition, namely 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 . 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 . This gives that the coefficient of g k in the product is P g i g j = g k α i β j . Notice,that while group rings can use rings and groups of arbitrary cardinality, werestrict ourselves to finite groups and finite rings. Note that we have notassumed that group nor the ring is commutative.The space of k by k matrices with coefficients in the ring R is denotedby M k ( R ) . It is immediate that M k ( R ) is a ring, however, it is, in general,a non-commutative ring. Moreover, it is fundamental in the study of non-commutative rings since any finite ring moded out by its Jacobson radicalis isomorphic to a direct product of matrix rings. Moreover, we know that M k ( R ) is a Frobenius ring, when R is Frobenius.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 =3 IRC ( α , α . . . , α n ) , where α i are ring elements. A block-circulant matrixis one where each row contains blocks which are square matrices. The rows ofthe block matrix are defined by shifting one block to the right relative to thepreceding row. We label the block-circulant matrix as CIRC( A , A , . . . A n ) , where A i are k × k matrices over the ring R. 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 north-east-to-south-west diagonal.
The following construction of a matrix which was used to construct codesthat were ideals in a group ring was first given for codes over fields by Hurleyin [12]. It was then extended to finite commutative Frobenius rings in [6].Let R be a finite commutative Frobenius ring and let G = { g , g , . . . , g n } bea group of order n . Let v = α g g + α g g + · · · + α g n g n ∈ RG.
Define thematrix σ ( v ) ∈ M n ( R ) to be σ ( v ) = α g − g α g − g α g − g . . . α g − g n α g − g α g − g α g − g . . . α g − g n ... ... ... ... ... α g − n g α g − n g α g − n g . . . α g − n g n . (2.1)We note that the elements g − , g − , . . . , g − n are the elements of the group G in a some given order.This matrix was used as a generator matrix for codes. The form of thematrix guaranteed that the resulting code would correspond to an ideal inthe group ring and thus have the group G as a subgroup of its automorphismgroup, that is the group G , acting on the coordinates, would leave the codefixed. The fundamental purpose of this was to construct codes that werenot found using more traditional construction techniques. It was shown in[6] that certain classical constructions would only produce a subset of allpossible codes (in this case self-dual codes) and would often miss codes thatwere of particular interest. With this in mind we are interested in expanding4hese kinds of constructions to enable us to find codes that would be missedwith other construction techniques.We now generalize the matrix construction just defined. Let R be afinite commutative ring and let G = { g , g , . . . , g n } be a group of order n .We note that no assumption about the groups commutativity is made. 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 matrixwith entries from the ring R. Define the block matrix σ k ( v ) ∈ ( M k ( R )) n tobe 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 . (2.2)We note that the element v is an element of the group matrix ring M k ( R ) G. Of course, this is the same construction as was previously givenfor group rings but we specify it here since we will use it in very differentways. Namely, we can consider the matrix as generating two distinct codesin different ambient spaces.This group matrix ring can be non-commutative in two ways. First, sincethe group may not be commutative, multiplication on the left by an element g ∈ G can give a different element than multiplication on the right by g . Wenote that in generating the matrix σ k ( v ) the rows are formed from elementsthat were constructed by a group element multiplying on the left. Moreover,multiplication by an element B ∈ M k ( R ) on the left can give a differentelement than multiplication on the right by B .As in the matrix σ ( v ) from Equation 2.1, the elements g − , g − , . . . , g − n are the elements of the group G given in a some order. This order is used inorder aid in the computational aspects of some proofs. We note that when k = 1 then σ ( v ) = σ ( v ) , that is, σ ( v ) is equivalent to the matrix σ ( v ) inthe original definition. In general, we shall often assume that k > . The next theorem sets up some useful algebraic tools.
Theorem 2.1.
Let R be a finite commutative ring. Let G be a group oforder n with a fixed listing of its elements. Then the map σ k : M k ( R ) G → M n ( M k ( R )) is a bijective matrix ring homomorphism.Proof. Let G = { g , g , . . . , g n } be the listing of the elements of G. Now definethe map σ k : M k ( R ) G → M n ( M k ( R )) as follows. Suppose v = P ni =1 A g i g i . 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 where each A g i is a square matrix of order k. It can be easily verified thatthis mapping is additive, surjective and injective. We now show that σ k ismultiplicative. Consider w = P ni =1 B g i g i then σ k ( w ) = B g − g B g − g B g − g . . . B g − g n B g − g B g − g B g − g . . . B g − g n ... ... ... ... ... B g − n g B g − n g B g − n g . . . B g − n g n . Now suppose w ∗ v = t, where t = P ni =1 C g i g i . Then σ k ( w ) ∗ σ k ( v ) = C g − g C g − g C g − g . . . C g − g n C g − g C g − g C g − g . . . C g − g n ... ... ... ... ... C g − n g C g − n g C g − n g . . . C g − n g n and this is σ k ( t ) = σ k ( w ∗ v ) as required.Let the first column of the matrix in Equation (2.2) be labelled by g , the second column by g , etc. Then if b = P ni =1 B g i g i is in M k ( R ) G thenthe coefficient of g i in the product b ∗ v is ( B g , B g , . . . , B g n ) times the i-thcolumn of σ k ( v ) . We note here that we are multiplying by b on the left. Thiscould easily be done on the right to get a similar result.We define the k by k matrix I k in the usual way. That is ( I k ) ij = 1 if i = j and ( I k ) ij = 0 if i = j. Since each ring we consider in this paper has amultiplicative identity this matrix is always an element in M k ( R ) . Theorem 2.2.
Let R be a finite commutative ring. Then v ∈ M k ( R ) G is aunit in M k ( R ) G if and only if σ k ( v ) is a unit in M n ( M k ( R )) . Proof.
Suppose v is a unit in M k ( R ) G and that w is its inverse. Then v ∗ w =( I k ) M k ( R ) G and hence σ k ( v ∗ w ) = σ k (( I k ) M k ( R ) G ) = I kn , the identity matrix6n M n ( M k ( R )) . Thus σ k ( v ) ∗ σ k ( w ) = I kn . Similarly, σ k ( w ) ∗ σ k ( v ) = I kn andso σ k ( v ) is invertible in M n ( M k ( R )) . Suppose now that σ k ( v ) is a unit in M n ( M k ( R )) and let N denote itsinverse. Let v = P ni =1 A g i g i . Then 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 where each A g i is a square matrix of order k. Let ( B , B , . . . , B n ) be the firstrow of N, where B i are the square matrices each of order k. Then: B A g − g + B A g − g + . . . + B n A g − n g = I k ,B A g − g + B A g − g + . . . + B n A g − n g = , ... ... ... ... ... ... ... ... ... B A g − g n + B A g − g n + . . . + B n A g − n g n = . (2.3)Now v = A g g + A g g + · · · + A g n g n = A g − i g g − i g + A g − i g g − i g + · · · + A g − i g n g − i g n , for each i, ≤ i ≤ n. Define w = B g + B g + · · · + B n g n . Then: B i g i ( A g g + A g g + · · · + A g n g n ) = B i g i A g − i g g − i g + B i g i A g − i g g − i g ++ · · · + B i g i A g − i g n g − i g n = B i A g − i g g + B i A g − i g g + · · · + B i A g − i g n g n . Hence: v ∗ w = ( B g + B g + · · · + B n g n )( A g g + A g g + · · · + A g n g n ) equalsto: B A g − g g + B A g − g g + . . . + B n A g − n g g + B A g − g g + B A g − g g + . . . + B n A g − n g g ... ... ... ... ... ... ... ...+ B A g − g n g n + B α g − g n g n + . . . + B n A g − n g n g n and this is g from the above. Thus g − ∗ w is the inverse of v and v is a unitin M k ( R ) . Group Matrix Ring Codes
In this section, we employ the matrix construction from the previous sectionto generate codes in two different ambient spaces. We make two distinctconstructions.
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 . (3.1)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 . (3.2)Here 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.The following lemma is immediate. Lemma 3.1.
Let R be a finite Frobenius ring and let G be a group of order n . Let v ∈ M k ( R ) G .1. The matrix σ k ( v ) is an n by n matrix with elements from M k ( R ) andthe code C k ( v ) is a length n code over M k ( R ) .2. The matrix τ k ( v ) is an nk by nk matrix with elements from R and thecode B k ( v ) is a length nk code over R . We illustrate these construction techniques in the following example.
Example . Let v = (cid:18) (cid:19) + (cid:18) (cid:19) a + (cid:18) (cid:19) a + (cid:18) (cid:19) a + (cid:18) (cid:19) b ++ (cid:18) (cid:19) ba + (cid:18) (cid:19) ba + (cid:18) (cid:19) ba ∈ M ( F ) D , h a, b i ∼ = D , the dihedral group with 8 elements. Then σ ( v )generates a code C ( v ) which is the ambient space M ( F ) . The matrix τ ( v ) = and τ ( v ) can be row reduced to . It can be easily checked that B ( v ) is a binary self-dual code with parameters[16 , , τ k ( v ) does not only depend9n the ring elements and the finite group G as does the matrix σ k ( v ), butrather τ k ( v ) also depends on the form of the matrices A g i . We note that the k × k matrices A g i over R can each take a different form - this is the firstadvantage of our generalization over the matrix σ ( v ). We also note thatthe matrix σ k ( v ) gives us more freedom for controlling the search field whenfinding a special family of codes since the matrices A g i do not have to befully defined by the ring elements appearing in their first rows - and this isthe second advantage of our generalization over the matrix σ ( v ) . Theorem 3.3.
Let R be a finite commutative Frobenius ring, k a positiveinteger and G a finite group of order n. Let v ∈ M k ( R ) G . Let I k ( v ) bethe set of elements of M k ( R ) G such that P A i g i ∈ I k ( v ) if and only if ( A , A , . . . , A n ) ∈ C k ( v ) . Then I k ( v ) is a one-sided ideal in M k ( R ) G andin particular, it is a left ideal.Proof. Each row of σ k ( v ) corresponds to an element of the form hv in M k ( R ) G, where h is any element of G. That is, the multiplication by h is done fromleft. The sum of any two elements in I ( v ) corresponds exactly to the sum ofthe corresponding elements in C k ( v ) and so I k ( v ) is closed under addition.Now we shall show when the product of an element in M k ( R ) G andan element in I k ( v ) is in I k ( v ) . Let w = P B i g i ∈ M k ( R ) G, where B i are the k × k matrices. Then if w is a row in C k ( v ) , it is of the form P C j h j v. Then w w = P B i g i P C j h j v = P B i C j g i h j v which correspondsto an element in C k ( v ) gives that the element is in I k ( v ) . Next, consider w w = P C j h j v P B i g i = P C j h j vB i g i which may not be an element in C k ( v ) . Thus, I k ( v ) is a left ideal of M k ( R ) G and since M k ( R ) is a non-commutative matrix ring, we have that I k ( v ) is a one-sided ideal of M k ( R ) G. Given this theorem, we know that any code C k ( v ) has G as a subgroupof its automorphism group. This is not true, of course, for the code B k ( v ).The above two results highlight the difference between group codes stud-ied in [6] and the codes we explore in this work. Namely, in group codes itis the coordinates that are held invariant by the action of the group G andin the codes we study in this work, it is the blocks that are held invariant bythe action of the group G. For this reason, from now on, we refer to codes C k ( v ) as group matrix ring codes.Now we show that the orthogonal of a group matrix ring code for somegroup G is also a group matrix ring code. Let J k be a one-sided, left ideal10n a group matrix ring M k ( R ) G. Define R ( C ) = { w | vw = 0 , ∀ v ∈ J k } . It isimmediate that R ( J k ) is a one-sided, right ideal of M k ( R ) G. Let v = A g g + A g g + · · · + A g n g n ∈ M k ( R ) G and C k ( v ) be the cor-responding group matrix ring code. Let Ψ : M k ( R ) G → ( M k ( R )) n be thecanonical map that sends A g g + A g g + · · · + A g n g n to ( A g , A g , . . . , A g n ) . Let J k be the one-sided, left ideal Ψ − ( C ) . Let w = ( B , B , . . . , B n ) ∈ R ( C ) . Then [( A g − j g , A g − j g , . . . , A g − j g n ) , ( B , B , . . . , B n )] = 0 , ∀ j. (3.3)This gives that n X i =1 A g − j g i B i = 0 , ∀ j. (3.4)Let w = Ψ − ( w ) = P B g i g i and define w ∈ M k ( R ) G to be w = C g g + C g g + · · · + C g n g n where C g i = B g − i . (3.5)Then n X i =1 A g − j g i B i = 0 = ⇒ n X i =1 A g − j g i C g − i = 0 . (3.6)Then g − j g i g − i = g − j , hence this is the coefficient of g − j in the product of w and g − j v. This gives that w ∈ R ( J k ) if and only if w ∈ R ( C ) . Let φ : ( M k ( R )) n → M k ( R ) G by φ ( w ) = w . It is clear that φ is a bijectionbetween R ( C ) and R (Ψ − ( C )) . Theorem 3.4.
Let C = C k ( v ) be a group matrix ring code in M k ( R ) formedfrom the element v ∈ M k ( R ) G. Then Ψ − ( R ( C )) is a one-sided, left idealof M k ( R ) G. Moreover, if C k ( v ) is a left-linear matrix ring G -code with theelements of the group acting on the left then R ( C k ( v )) is a right-linear matrixgroup G -code with the elements of the group acting on the right.Proof. Follows from the above discussion.We can now investigate the situation for the code B k ( v ) . We begin witha definition. Let G be a finite group of order n and R a finite Frobenius11ommutative ring. Let D be a code in R sn where the coordinates can bepartitioned into n sets of size s where each set is assigned an element of G . Ifthe code D is held invariant by the action of multiplying the coordinate setmarker by every element of G then the code D is called a quasi-group codeof index s . Lemma 3.5.
Let R be a finite Frobenius ring and let G be a finite groupwith v ∈ M k ( R ) . Then B k ( v ) is a quasi- G -code of length nk and index k .Proof. Let v ∈ M k ( R ) G and let w be a row of the matrix τ k ( v ). Letting anyelement in G act on the k coordinates corresponding to the matrices, gives anew row of τ k ( v ) . Therefore, the code B k ( v ) is a quasi- G -code of length nk and index k .Consider a quasi- G -code of index k . Then rearranging the coordinatesso that the i -th coordinates of each group of k coordinates are placed se-quentially, then it is easy to see that any ( g , g , . . . , g n ) ∈ G n holds the codeinvariant. Namely, any quasi- G -code of length kn and index k is a G k -code.This gives the following. Theorem 3.6.
Let R be a finite Frobenius ring and let G be a finite groupwith v ∈ M k ( R ) . Then B k ( v ) is a G k code of length kn. Proof.
Follows from Lemma 3.5 and the previous discussion. [ I kn | τ k ( v )] and Self-Dual Codes In this section, we investigate constructions of binary self-dual codes from τ k ( v ) . Lemma 4.1.
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 .Proof. Follows from the standard proof that ( I m | A ) generates a self-dualcode of length 2 m if and only if AA T = − I m .12ecall 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 . (4.1)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 . (4.2) Lemma 4.2.
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.Proof. The proof is similar to the proof in Theorem 2.1 and simply consistsof showing that addition and multiplication are preserved.Now, combining together Lemma 4.1, Lemma 4.2 and the fact that τ k ( v ) = − I kn if and only if v = − I k , we get the following corollary. Corollary 4.3.
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: Corollary 4.4.
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 search for binary self-dual codes with parameters [72 , , I kn | τ kn ( v )] , where v ∈ M k ( F ) G k and different groups G to show the strength of ourconstruction and particularly, the strength of the matrix τ kn ( v ) . The possible weight enumerators for a Type I [72 , ,
12] codes are asfollows ([7]): 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 ([7]):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, 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 into subsections, where in eachwe consider a generator matrix of the form [ I kn | τ k ( v )] for a specific group G and some specific k × k block matrices to search for binary self-dual codeswith parameters [72 , , . All the upcoming computational results wereobtained by performing searches using a particular algorithm technique (see[15] for details) in the software package MAGMA ([1]). C and × block matrices In this section, we consider the cyclic group C with some 18 ×
18 blockmatrices. Let G = h x | x = 1 i ∼ = C . Let v = P i =0 Y i x i ∈ ( M ( F )) C , then τ ( v ) = (cid:18) Y Y Y Y (cid:19) , (4.3)where Y = (cid:18) A BB A (cid:19) , Y = (cid:18) C DD C (cid:19) with A = CIRC ( A , A , A ) ,B = CIRC ( A , A , A ) , = CIRC ( A , A , A ) ,D = CIRC ( A , A , A )where A i are some matrices. We now employ a generator matrix of the form[ I | τ ( v )] , where I is the 36 ×
36 identity matrix, for different forms of thematrices A i to search for binary self-dual codes with parameters [72 , , . We only list codes with parameters in their weight distributions that werenot known in the literature before. Also, since the matrix σ ( v ) is fullydefined by the first row, we only list the first row of the matrices Y , and Y which we label as r Y and r Y respectively.Case 1. Here we let A = revcirc ( a , a , a ) ,A = revcirc ( a , a , a ) ,. . . ,A = revcirc ( a , a , a ) . Table 1: New Type I [72 , ,
12] Codes
Type r Y r Y γ β | Aut ( C i ) | C W , (0 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 36 543 72
Table 2: New Type II [72 , ,
12] Codes r Y r Y α | Aut ( C i ) | C (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , , − Case 2. Here we let A = revcirc ( a , a , a ) ,A = revcirc ( a , a , a ) ,. . . ,A = revcirc ( a , a , a ) ,A = circ ( a , a , a ) ,A = circ ( a , a , a ) ,. . . ,A = circ ( a , a , a ) . , ,
12] Codes
Type r Y r Y γ β | Aut ( C i ) | C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 0 342 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 420 36 C W , (0 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , ,
1) 36 561 72
Table 4: New Type II [72 , ,
12] Codes r Y r Y α | Aut ( C i ) | C (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , , − Case 3. Here we let A = circ ( a , a , a ) ,A = circ ( a , a , a ) ,. . . ,A = circ ( a , a , a ) ,A = revcirc ( a , a , a ) ,A = revcirc ( a , a , a ) ,. . . ,A = revcirc ( a , a , a ) . Table 5: New Type I [72 , ,
12] Codes
Type r Y r Y γ β | Aut ( C i ) | C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 0 186 36 C W , (0 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
1) 18 432 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 36 597 72
Table 6: New Type II [72 , ,
12] Codes r Y r Y α | Aut ( C i ) | C (1 , , , , , , , , , , , , , , , , ,
0) (1 , , , , , , , , , , , , , , , , , − C (1 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , , − .1.2 The group D and × block matrices In this section, we consider the dihedral group D with some 2 × G = h x, y | x = y = 1 , x y = x − i ∼ = D . Let v = P i =0 P j =0 A i +9 j x i y j ∈ M ( F ) D , then τ ( v ) = (cid:18) A BB T A T (cid:19) , (4.4)with A = CIRC ( A , A , A , . . . , A ) ,B = CIRC ( A , A , A , . . . , A )where A i are some matrices.We now employ a generator matrix of the form [ I | τ ( v )] , where I is the36 ×
36 identity matrix, for different forms of the matrices A i to search forbinary self-dual codes with parameters [72 , , . We only list codes withparameters in their weight distributions that were not known in the literaturebefore.Case 1. Here we let A = circ ( a , a ) ,A = circ ( a , a ) ,. . . ,A = circ ( a , a ) . Since τ ( v ) is fully defined by the first row, we only list the first rowsof the matrices A and B which we label as r A and r B respectively.Table 7: New Type I [72 , ,
12] Codes
Type r A r B γ β | Aut ( C i ) | C W , (0 , , , , , , , , , , , , , , , , ,
1) (0 , , , , , , , , , , , , , , , , ,
0) 18 237 36 C W , (1 , , , , , , , , , , , , , , , , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 18 387 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 36 417 36 C W , (1 , , , , , , , , , , , , , , , , ,
0) (0 , , , , , , , , , , , , , , , , ,
0) 36 564 36
Case 2. Here we let A = (cid:18) a a a a (cid:19) , A = (cid:18) a a a a (cid:19) , A = (cid:18) a a a a (cid:19) , . . . , A = (cid:18) a , a a , a (cid:19) A = circ ( a , a ) ,A = circ ( a , a ) ,A = circ ( a , a ) ,. . . ,A = circ ( a , a ) . We note here, that the first nine blocks are the 2 × × . To save space, we only list the three variables of each persymmetricmatrix which we label as r A , r A , r A , . . . , r A and the first row of thematrix B which we label as r B since this matrix is fully defined by thefirst row. Table 8: New Type I [72 , ,
12] Codes
Type r A r A r A r A r A r A r A r A r A r B γ β | Aut ( C i ) | C W , (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
1) (0 , ,
1) (0 , ,
1) (1 , ,
1) (1 , ,
0) (0 , , , , , , , , , , , , , , , , ,
1) 9 264 18 C W , (0 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
1) (0 , ,
0) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
1) (1 , , , , , , , , , , , , , , , , ,
0) 27 345 18
We note that the code C in the above table is the first example of aself-dual [72 , ,
12] code with γ = 27 in its weight distribution. C , and × block matrices In this section, we consider the cyclic group C , and some 2 × G = h x | x · = 1 i ∼ = C , . Let v = P i =0 P j =0 A i +6 j x i + j ∈ M ( F ) C , , then τ ( v ) = A B CC ′ A BB ′ C ′ A , (4.5)where A = CIRC ( A , A , . . . , A ) ,B = CIRC ( A , A , . . . , A ) ,C = CIRC ( A , A , . . . , A ) , ′ = CIRC ( A , A , A , . . . , A ) ,C ′ = CIRC ( A , A , A , . . . , A )and where A i are some 2 × I | τ ( v )] , where I is the36 ×
36 identity matrix, for different forms of the matrices A i to search forbinary self-dual codes with parameters [72 , , . We only list codes withparameters in their weight distributions that were not known in the literaturebefore.Case 1. Here we let A = circ ( a , a ) ,A = circ ( a , a ) ,A = circ ( a , a ) ,. . . ,A = circ ( a , a ) . We note that the search field here is 2 . Since the matrix τ ( v ) is fullydefined by the first row, we only list the first rows of the matrices A, B and C which we label as r A , r B and r C respectively.Table 9: New Type I [72 , ,
12] Codes
Type r A r B r C γ β | Aut ( C i ) | C W , (1 , , , , , , , , , , ,
0) (1 , , , , , , , , , , ,
0) (1 , , , , , , , , , , ,
1) 36 423 72
Case 2. Here we let A = (cid:18) a a a a (cid:19) , A = (cid:18) a a a a (cid:19) , A = (cid:18) a a a a (cid:19) , . . . , A = (cid:18) a , a a , a (cid:19) and A = circ ( a , a ) ,A = circ ( a , a ) ,A = circ ( a , a ) ,. . . ,A = circ ( a , a ) .
19e note here, that the first nine blocks are the 2 × × .To save space, we only list the three variables of each persymmetric ma-trix which we label as r A , r A , r A , . . . , r A and the first rows of the ma-trices A , A , A , . . . , A which we label as r A , r A , r A , . . . , r A since these matrices are each defined by the first row.Table 10: New Type I [72 , ,
12] Codes
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) (0 , ,
0) (0 , ,
1) (1 , ,
0) (1 , ,
0) (0 , ,
1) (0 , ,
1) (1 , ,
1) (1 , ,
1) 18 342 36 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 ,
1) (0 ,
0) (0 ,
1) (0 ,
0) (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 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
1) (0 , ,
0) (0 , ,
0) 36 510 36 r A r A r A r A r A r A r A r A r A (1 ,
1) (1 ,
1) (1 ,
0) (0 ,
1) (0 ,
0) (1 ,
0) (0 ,
0) (1 ,
1) (1 , We would like to stress that the above constructions represent a verysmall fraction of the possible matrix constructions that can be derived for thegenerator matrix [ I kn | τ k ( v )] . That is, there are many more different choicesfor the groups and their sizes, the forms of the k × k matrices and their sizeswhich can all lead to constructing optimal binary self-dual codes of variouslengths - this shows the strength of our generator matrix. In this work, we defined group matrix ring codes that are left ideals in thegroup matrix ring M k ( R ) G. We generalized a well known matrix construc-tion so that this generalization can be used to generate codes in two differentambient spaces. We presented a generator matrix for self-dual codes whichwe believe can be used to construct many new codes that could not be ob-tained from other, known in the literature, generator matrices. Additionally,we employed our generator matrix to search for binary self-dual codes. Inparticular, we constructed Type I binary [72 , ,
12] self-dual codes with newweight enumerators in W , : 20 γ = 0 , β = { , } ) , ( γ = 9 , β = { } ) , ( γ = 18 , β = { , , , , } ) , ( γ = 27 , β = { } ) , ( γ = 36 , β = { , , , , , , } )and Type II binary [72 , ,
12] self-dual codes with new weight enumerators:( α = {− , − , − , − } ) . A suggestion for future work is to consider the generator matrix we pre-sented in this work, for different groups, different types of the k × k matricesand different alphabets to search for new optimal binary self-dual codes ofdifferent lengths. 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.[2] 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 , , , ,
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.[13] 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