Quaternary Hermitian self-dual codes of lengths 26, 32, 36, 38 and 40 from modifications of well-known circulant constructions
aa r X i v : . [ m a t h . C O ] F e b QUATERNARY HERMITIAN SELF-DUAL CODES OF LENGTHS 26, 32, 36, 38AND 40 FROM MODIFICATIONS OF WELL-KNOWN CIRCULANTCONSTRUCTIONS
A. M. ROBERTS
Abstract.
In this work, we give three new techniques for constructing Hermitian self-dual codesover commutative Frobenius rings with a non-trivial involutory automorphism using λ -circulantmatrices. The new constructions are derived as modifications of various well-known circulantconstructions of self-dual codes. Applying these constructions together with the building-up con-struction, we construct many new best known quaternary Hermitian self-dual codes of lengths 26,32, 36, 38 and 40. Introduction
Hermitian self-dual codes form a class of linear codes which are self-dual with respect to theHermitian inner product. Let d ( n ) be the minimum distance of a quaternary Hermitian self-dualcode of length n . An upper bound on d ( n ) was given in [24] as d ( n ) ≤ ⌊ n/ ⌋ + 2 . A Hermitian self-dual code whose minimum distance meets its corresponding bound is called extremal . A Hermitian self-dual code with the highest possible minimum distance for its length issaid to be optimal . Extremal codes are necessarily optimal but optimal codes are not necessarilyextremal. A best known
Hermitian self-dual code is a Hermitian self-dual code with the highestknown minimum distance for its length.The existence of an extremal quaternary Hermitian self-dual code for lengths greater than 30 isstill an open problem. It was proved in [26] that there exists no extremal quaternary Hermitian self-dual code of length 26. A complete classification of quaternary Hermitian self-dual codes for lengthsup to 22 is given in [17]. Optimal quaternary Hermitian self-dual codes of length 24 possessingnon-trivial automorphisms of order ≥ n × n matrices touse in constructions of (Hermitian) self-dual codes, by assuming these matrices are circulant wereduce the size of the search field from n to n . For this reason, circulant matrices have beenused extensively to construct (Hermitian) self-dual codes. See [11, 7, 12, 2, 13, 10, 9] for recentutilisation of circulant matrices in constructing self-dual codes. In this work, we give three differentmodifications of various well-known circulant constructions of self-dual codes, which we apply toconstruct optimal and best known quaternary Hermitian self-dual codes. All of the new techniquescan be used to construct Hermitian self-dual codes over any commutative Frobenius ring R with afixed non-trivial involutory automorphism : R → R . We introduce these techniques and providethe conditions needed to produce a Hermitian self-dual code. Key words and phrases.
Hermitian self-dual codes, codes over rings, λ -circulant matrix, optimal codes, best knowncodes. For the proofs of these techniques, we utilise a specialised mapping Θ which was used in [27, 14].This mapping is inherently associated with the matrix product BA T , where A and B are λ -circulantmatrices over R such that λλ = 1. If A is the λ -circulant matrix generated by a ∈ R n , then usingΘ allows us to verify the equality AA T = − I n by computing the values of ⌊ n/ ⌋ + 1 quantities interms of a . This eliminates the need to construct A from its generating vector as well as computingthe matrix product AA T itself, which improves computational efficiency. We give and prove ourown results concerning Θ as done so in [27, 14] and we further generalise them with respect to theHermitian inner product.Using the new techniques together with the building-up construction, we find many Hermitian self-dual codes with weight enumerator parameters of previously unknown values (relative to referencedsources). In total, 408 new codes are found, including •
71 quaternary Hermitian self-dual [26 , , •
82 quaternary Hermitian self-dual [32 , , • , , • , , •
252 quaternary Hermitian self-dual [40 , , Preliminaries
Let R be a commutative Frobenius ring with a fixed non-trivial involutory automorphism : R → R (see [5] for a full description of Frobenius rings andcodes over Frobenius rings). Throughout this work, we always assume R has unity. A code C oflength n over R is a subset of R n whose elements are called codewords. If C is a submodule of R n ,then we say that C is linear. Let x , y ∈ R n where x = ( x , x , . . . , x n ) and y = ( y , y , . . . , y n ). TheHermitian dual C ⊥ H of C is given by C ⊥ H = { x ∈ R n : h x , y i H = 0 , ∀ y ∈ C} , where h , i H denotes the Hermitian inner product defined by h x , y i H = n X i =1 x i y i . We say that C is Hermitian self-orthogonal if C ⊆ C ⊥ H and Hermitian self-dual if C = C ⊥ H .If R = F p m for some prime p and m ∈ N , then we define the involutory automorphism a = a p m , ∀ a ∈ R . We can extend this to the finite commutative ring R = F p m + u F p m where F p m + u F p m = { a + bu : a, b ∈ F p m , u = 0 } and we define a + bu = a p m + b p m u , ∀ a, b ∈ R .Two codes C and C ′ over R are said to be conjugation equivalent or simply equivalent if there existsa monomial matrix M over R and an automorphism ν of R such that C ′ = ν ( C M ) = { ν ( c M ) : c ∈ C} .If C = ν ( C M ), then M and ν are said to form an automorphism of C . The set of all automorphismsof C forms the automorphism group Aut( C ) of C . In this paper, we consider the alphabets F and F + u F .We define F ∼ = F [ ω ] / h ω + ω + 1 i so that F = { aω + b (1 + ω ) : a, b ∈ F , ω + ω + 1 = 0 } . Define F + u F = { a + bu : a, b ∈ F , u = 0 } . Then F + u F is a commutative ring of order 16 and characteristic 2 such that F + u F ∼ = F [ u ] / h u i ∼ = F [ ω, u ] / h ω + ω + 1 , u , ωu + uω i .We recall the following Gray map from [23]: ϕ F + u F : ( F + u F ) n → F n ,a + bu ( b, a + b ) , a, b ∈ F n . It was shown in [21] that if C is a Hermitian self-dual code over F + u F of length n , then ϕ F + u F ( C ) is a Hermitian self-dual code over F of length 2 n . The Lee weight of a vector x ∈ ( F + u F ) n can be defined to be w L ( x ) = n ( x ) + 2 n ( x ) where n ( x ) is the number of componentsof x equal to a + bu with a = b or b = 0 and n ( x ) is the number of components of x equal to a + bu with a = b and b = 0. It is true that ϕ F + u F is an isometry from ( F + u F ) n under Leedistance to F n under Hamming distance. In this way, the minimum Lee distance and Lee weightenumerator of a code C over F + u F are equal to the minimum Hamming distance and Hammingweight enumerator of ϕ F + u F ( C ), respectively. We now recall the definitions and properties of some special matriceswhich we use in our work. Let a = ( a , a , . . . , a n − ) ∈ R n where R is a commutative ring and let A = a a a · · · a n − λa n − a a · · · a n − λa n − λa n − a · · · a n − ... ... ... . . . ... λa λa λa · · · a , where λ ∈ R . Then A is called the λ -circulant matrix generated a , denoted by A = circ λ ( a ). If λ = 1,then A is called the circulant matrix generated by a and is more simply denoted by A = circ( a ). Ifwe define the matrix P λ = (cid:16) I n − λ (cid:17) , then it follows that A = P n − i =0 a i P iλ . Clearly, the sum of any two λ -circulant matrices is also a λ -circulant matrix. If B = circ λ ( b ) where b = ( b , b , . . . , b n − ) ∈ R n , then AB = P n − i =0 P n − j =0 a i b j P i + jλ .Since P nλ = λI n there exist c k ∈ R such that AB = P n − k =0 c k P kλ so that AB is also λ -circulant. Infact, it is true that c k = X [ i + j ] n = ki + j Let J n be an n × n matrix over R whose ( i, j ) th entry is 1 if i + j = n + 1 and 0 if otherwise. Then J n is called the n × n exchange matrix and corresponds to the row-reversed (or column-reversed)version of I n . We see that J n is both symmetric and involutory, i.e. J n = J Tn and J n = I n . For anymatrix A ∈ R m × n , premultiplying A by J m and postmultiplying A by J n inverts the order in whichthe rows and columns of A appear, respectively. Namely, the ( i, j ) th entries of J m A and AJ n arethe ([1 − i ] m , j ) th and ( i, [1 − j ] n ) th entries of A , respectively. Note that [ i + j ] n corresponds to the( i, j ) th entry of the matrix J n V where V = circ((0 , , , . . ., n − We now introduce and explore the properties of a mapping which wasused in [27, 14]. The mapping is inherently associated with the matrix product BA T , where A and B are λ -circulant matrices such that λλ = 1. By utilising Θ, we are able to improve the computationalefficiency of our algorithms. Definition 2.1. ([27, 14]) Let R be a commutative ring and let n ∈ N be fixed. Let Θ : R n × R n × Z n → R be a mapping with an optional argument λ ∈ R defined by Θ( x , y , j )[ λ ] = n − j − X i =0 x [ i + j ] n y i + λ n − X i = n − j x [ i + j ] n y i , where x = ( x , x , . . . , x n − ) , y = ( y , y , . . . , y n − ) ∈ R n and j ∈ [0 .. n − .If j = 0 , we define Θ( x , y , 0) = n − X i =0 x i y i = xy T , which is independent of λ .If λ is unspecified, then we assume λ = 1 so that Θ( x , y , j ) = n − X i =0 x [ i + j ] n y i . Lemma 2.2. ([27]) Let R be a commutative ring with a fixed non-trivial involutory automorphism : R → R . Let x , y ∈ R n and let λ ∈ R : λλ = 1 . Then Θ( x , y , j )[ λ ] = λ Θ( y , x , n − j )[ λ ] , ∀ j ∈ [0 .. n − .Proof. If x = ( x , x , . . . , x n − ), then x [ i + k ] n = ˜ x s , where ˜ x is the vector x after being circularlyshifted by k places for some k ∈ [0 .. n − y = ( y , y , . . . , y n − ), then in a correspondencebetween the elements x i and y i , inflicting a circular shift to both x and y by the same number ofplaces preserves this correspondence. Thus, noting that λλ = 1 by assumption, we have λ Θ( y , x , n − j )[ λ ] = λ n − ( n − j ) − X i =0 y [ i +( n − j )] n x i + λ n − X i = n − ( n − j ) y [ i +( n − j )] n x i = λ j − X i =0 y [ i +( n − j )] n x i + n − X i = j y [ i +( n − j )] n x i = n − X i = j x [ i + j +( n − j )] n y [ i +( n − j )] n + λ j − X i =0 x [ i + j +( n − j )] n y [ i +( n − j )] n = n − j − X i =0 x [ i + j ] n y i + λ n − X i = n − j x [ i + j ] n y i = Θ( x , y , j )[ λ ] . (cid:3) Remark . In Lemma 2.2, suppose we want to calculate f ( j ) = Θ( x , x , j )[ λ ], ∀ j ∈ [0 .. n − f ( j ) = λ Θ( x , x , n − j )[ λ ] which, since λλ = 1, implies f ( n − j ) = Θ( x , x , n − j )[ λ ] = λ Θ( x , x , j )[ λ ] = λf ( j )so that f ( j ) = λ − f ( n − j ) = λf ( n − j ). Therefore, to calculate f ( j ) for j ∈ [0 .. n − f ( j ) for j ∈ [0 .. ⌊ n/ ⌋ ].Likewise, let a i ∈ R n for i ∈ [0 .. k − 1] and suppose we want to calculate g ( j, i, t ) = Θ( a [ i + j ] k , a i , t )[ λ ], ∀ j ∈ [1 .. k − 1] and t ∈ [1 .. n − g ( j, i, t ) = Θ( a [ i + j ] k , a i , t )[ λ ]= λ Θ( a i , a [ i + j ] k , n − t )[ λ ]= λ Θ( a i , a [ i + j ] k , n − t )[ λ ]= λ Θ( a [( i + j )+( k − j )] k , a [ i + j ] k , n − t )[ λ ]= λg ([ i + j ] k , k − j, n − t ) . Let G t be the matrix whose ( i, j ) th entry is g ( j, i, t ) for fixed t ∈ [1 .. n − T be thetransformation which multiplies each entry of a matrix by λ and let T be the transformationwhich circularly shifts the j th row of a matrix to the right by j places, ∀ j . Then we see that G t = T ( J k − T ( G n − t )), ∀ t , where J k − is the ( k − × ( k − 1) exchange matrix. Conversely, since T and T are clearly both invertible, we have G n − t = T − ( J k − T − ( G t )). Therefore, to calculate g ( j, i, t ) for j ∈ [1 .. k − 1] and t ∈ [1 .. n − g ( j, i, t ) for j ∈ [1 .. k − t ∈ [1 .. ⌊ n/ ⌋ ].Finally, suppose we want to calculate g ( j, i, 0) = Θ( a [ i + j ] k , a i , ∀ j ∈ [1 .. k − g ( j, i, 0) = g ([ i + j ] k , k − j, v j be the vector whose i th entry is g ( j, i, v j corresponds to the vector v k − j after be circularly shifted to the leftby j places. Therefore, to calculate g ( j, i, 0) for j ∈ [1 .. k − g ( j, i, j ∈ [1 .. ⌊ k/ ⌋ ]. Lemma 2.4. ([27]) Let R be a commutative ring with a fixed non-trivial involutory automorphism : R → R . Let A = circ λ ( a ) and B = circ λ ( b ) with a , b ∈ R n and λ ∈ R : λλ = 1 . Then BA T = circ λ (( v , v , . . ., v n − )) , where v j = Θ( b , a , j )[ λ ] , ∀ j ∈ [0 .. n − .Proof. Since λλ = 1 by assumption, we know that BA T is λ -circulant such that BA T = circ λ (( x y T , x y T , . . ., x y nT )), where x i and y i denote the i th rows of B and A , respectively. Let BA T =circ λ (( v , v , . . ., v n − )) so that v j = x y j +1 T for j ∈ [0 .. n − v = P n − i =0 b i a i .In the product v = x y T , we see that the indices of the vector y = ( λa n − , a , a , . . . , a n − ) cor-respond to the indices of the vector x = ( b , b , b , . . . , b n − ) after being circularly shifted to theright by 1 place. Thus, in v = x y T , there is a summation of terms in the form b [ i +1] n a i for i ∈ [0 .. n − x y j +1 T , there is a summa-tion of terms in the form b [ i + j ] n a i for i ∈ [0 .. n − 1] and j ∈ [1 .. n − v , we see that theterms of the summation will acquire λ as a coefficient for i = n − 1. By extending this argument, in v j , we see that the terms of the summation will acquire λ as a coefficient for i ∈ [ n − j .. n − 1] and j ∈ [1 .. n − v j = (P n − i =0 b i a i , j = 0 , P n − j − i =0 b [ i + j ] n a i + λ P n − i = n − j b [ i + j ] n a i , j ∈ [1 .. n − v j = Θ( b , a , j )[ λ ], ∀ j ∈ [0 .. n − (cid:3) With these lemmas established, we will now look at how Θ can be used to prove that a matrix G = ( I n , A ) is a generator matrix of a Hermitian self-dual code. A. M. ROBERTS Proposition 2.5. ([27]) Let R be a commutative ring with a fixed non-trivial involutory automor-phism : R → R . Let A = circ λ ( a ) with a ∈ R n and λ ∈ R : λλ = 1 . Then AA T = − I n if and onlyif Θ( a , a , j )[ λ ] = (cid:26) − , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] . Proof. Since λλ = 1 by assumption, by Lemma 2.4 we have AA T = circ λ (( v , v , . . ., v n − )) where v j = Θ( a , a , j )[ λ ], ∀ j ∈ [0 .. n − AA T are given by v and v j , respectively, for j = 0, so AA T = − I n if and only if v = − v j = 0, ∀ j ∈ [1 .. n − v = − v j = 0, ∀ j ∈ [1 .. ⌊ n/ ⌋ ]. Therefore,we see that AA T = − I n if and only ifΘ( a , a , j )[ λ ] = (cid:26) − , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] . (cid:3) For example, if R is a commutative Frobenius ring with a fixed non-trivial involutory automor-phism : R → R such that λ ∈ R : λλ = 1, then in the pure double circulant construction ofHermitian self-dual codes given by G = ( I n , A ) for an n × n λ -circulant matrix A = circ λ ( a ) over R , we know that G is a generator matrix of a Hermitian self-dual [2 n, n ]-code over R if and only if AA T = − I n . In terms of Θ, by Proposition 2.5 this is true if and only ifΘ( a , a , j )[ λ ] = (cid:26) − , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] . The Constructions In this section, we present the three techniques for constructing Hermitian self-dual codes, allof which are derived as modifications of previously known constructions of self-dual codes. Wewill hereafter always assume R is a commutative Frobenius ring with a fixed non-trivial involutoryautomorphism : R → R , which we refer to as the Hermitian involution. The first technique we look at can be used to construct Hermitian self-dual [4 n, n ]-codes over R . It can be interpreted as a Hermitian modification of the four circulanttechnique for constructing self-dual codes first introduced in [1], which uses a matrix G defined by G = ( I n X ) , where X = (cid:18) A B − B T A T (cid:19) , and where A and B are circulant matrices. It also corresponds to the Hermitian analogue of theconstruction presented in [14]. Theorem 3.1. Let G = ( I n X ) , where X = (cid:18) − A T CJ − BB T CJ − A (cid:19) and where J = J n , A = circ λ ( a ) , B = circ λ ( b ) and C = circ µ ( c ) with a , b , c ∈ R n and λ, µ ∈ R : λλ = µµ = 1 . Then G is a generator matrix of a Hermitian self-dual [4 n, n ] -code over R if andonly if X x ∈ S Θ( x , x , j )[ λ ] = (cid:26) − , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] , Θ( c , c , j )[ µ ] = (cid:26) , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] , where S = { a , b } . Proof. We know that G is a generator matrix of a Hermitian self-dual [4 n, n ]-code over R if andonly if XX T = − I n . Since λλ = 1 by assumption, we have that A and B as well as their Hermitianinvolutions and transpositions all commute with one another multiplicatively. Firstly, since J issymmetric we have X T = (cid:18) − A T CJ − BB T CJ − A (cid:19) T = (cid:18) − JC T A JC T B − B T − A T (cid:19) . If the ( i, j ) th block-wise entry of XX T is x i,j , noting that J is involutory we see that x , = A T CJ C T A + BB T = A T CC T A + BB T ,x , = − A T CJ C T B + BA T = − A T CC T B + BA T ,x , = − B T CJ C T A + AB T = − B T CC T A + AB T ,x , = B T CJ C T B T + AA T = B T CC T B T + AA T . Noting that XX T = − I n if and only if A T CC T B = BA T , we see that XX T = − I n ⇐⇒ A T CC T A + BB T = − I n ⇐⇒ A T CC T AA T + ( BA T ) B T = − A T ⇐⇒ A T CC T AA T + ( A T CC T B ) B T = − A T ⇐⇒ A T CC T ( AA T + BB T ) = − A T ⇐⇒ CC T ( AA T + BB T ) = − I n and so, combined with our other required conditions, we have that XX T = − I n if and only if A T CC T A + BB T = − I n ,B T CC T B + AA T = − I n ,CC T ( AA T + BB T ) = − I n . Clearly, we must have CC T = I n for all of these equations to be satisfied. With this prerequisite,our conditions reduce to x , = AA T + BB T ,x , = ,x , = ,x , = AA T + BB T , or equivalently x , = AA T + BB T ,x , = ,x , = ,x , = AA T + BB T , Therefore, XX T = − I n if and only if AA T + BB T = − I n and CC T = I n and by Proposition2.5 this is true if and only if X x ∈ S Θ( x , x , j )[ λ ] = (cid:26) − , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] , Θ( c , c , j )[ µ ] = (cid:26) , j = 0 , , j ∈ [1 .. ⌊ n/ ⌋ ] , where S = { a , b } . (cid:3) A. M. ROBERTS Remark . Let U ′ denote the set of unitary elements in R , i.e. U ′ = { λ ∈ R : λλ = 1 } . Let N C = N C ( R, n ) denote the number of unitary µ -circulant matrices C (i.e. CC T = I n ) over R for all µ ∈ U ′ . The search field for Hermitian self-dual [4 n, n ]-codes over R constructed by Theorem 3.1is of size | R | n · | U ′ | · N C . In general, N C is relatively small, for example N C ( F , 10) = 4 , 320 and N C ( F + u F , 5) = 8 , Remark . In Theorem 3.1, we are in fact able to assume C is any matrix over R such that C isunitary. Moreover, C and C T need not commute multiplicatively with either A , B , their Hermitianinvolutions or their transpositions. The second technique can be used to construct Hermitian self-dual [2 kn, kn ]-codes over R where k ∈ N . It is derived as the Hermitian analogue of the technique for constructingself-dual codes first given in [8], which uses a matrix G defined by G = ( I kn X ) , where X = circ(( A , A , . . ., A k − )) , and where A i are circulant matrices for i ∈ [0 .. k − Theorem 3.4. Let G = ( I kn X ) , where X = circ λ (( A , A , . . ., A k − )) and where A i = circ µ ( a i ) with a i ∈ R n for i ∈ [0 .. k − and λ, µ ∈ R : λλ = µµ = 1 . Then G is agenerator matrix of a Hermitian self-dual [2 kn, kn ] -code over R if and only if k − X i =0 Θ( a i , a i , t )[ µ ] = (cid:26) − , t = 00 , t ∈ [1 .. ⌊ n/ ⌋ ] , k − j − X i =0 Θ( a [ i + j ] k , a i , 0) + λ k − X i = k − j Θ( a [ i + j ] k , a i , 0) = 0 , j ∈ [1 .. ⌊ k/ ⌋ ] , k − j − X i =0 Θ( a [ i + j ] k , a i , t )[ µ ] + λ k − X i = k − j Θ( a [ i + j ] k , a i , t )[ µ ] = 0 , j ∈ [1 .. k − ,t ∈ [1 .. ⌊ n/ ⌋ ] . Proof. We know that G is a generator matrix of a Hermitian self-dual [2 kn, kn ]-code over R if andonly if XX T = − I kn , namely the main diagonal and off-diagonal block-wise entries of XX T areequal to − I n and , respectively. Since λλ = 1 by assumption, we have that XX T is λ -circulantsuch that XX T = circ λ (( R R T , R R T , . . ., R R nT )), where R i denotes the i th block row of X .Following an argument similar to that used in the proof of Lemma 2.4, we observe that R R jT = (P k − i =0 A i A iT , j = 1 , P k − i =0 A [ i +( j − k A iT + λ P k − i = k − j +1 A [ i +( j − k A iT , j ∈ [2 .. k ] . (3.1)The main diagonal and off-diagonal block-wise entries of XX T are equal to R R T and R R jT for j ∈ [2 .. k ], respectively. Thus, by (3.1) we see that XX T = − I kn if and only if k − X i =0 A i A iT = − I n (3.2)and k − j − X i =0 A [ i + j ] k A iT + λ k − X i = k − j A [ i + j ] k A iT = , (3.3) ∀ j ∈ [1 .. k − µµ = 1, by Proposition 2.5 we see that (3.2) is satisfied if and only if k − X i =0 Θ( a i , a i , t )[ µ ] = (cid:26) − , t = 0 , , t ∈ [1 .. ⌊ n/ ⌋ ] . By Lemma 2.4, we see that A [ i + j ] k A iT = circ λ (( v j, , v j, , . . ., v j,n − )) where v j,t = Θ( a [ i + j ] k , a i , t )[ µ ], ∀ j ∈ [1 .. k − 1] and t ∈ [0 .. n − v j,t , ∀ j ∈ [1 .. k − 1] and t ∈ [1 .. ⌊ n/ ⌋ ] and v j, for j ∈ [1 .. ⌊ k/ ⌋ ]. Therefore, we find that (3.3) issatisfied if and only if k − j − X i =0 Θ( a [ i + j ] k , a i , 0) + λ k − X i = k − j Θ( a [ i + j ] k , a i , 0) = 0 , ∀ j ∈ [1 .. ⌊ k/ ⌋ and k − j − X i =0 Θ( a [ i + j ] k , a i , t )[ µ ] + λ k − X i = k − j Θ( a [ i + j ] k , a i , t )[ µ ] = 0 , ∀ j ∈ [1 .. k − 1] and t ∈ [1 .. ⌊ n/ ⌋ ]. (cid:3) Remark . Let U ′ denote the set of unitary elements in R , i.e. U ′ = { λ ∈ R : λλ = 1 } . The searchfield for Hermitian self-dual [2 kn, kn ]-codes over R constructed by Theorem 3.4 is of size | R | kn ·| U ′ | . The third technique can be used to construct Hermitian self-dual [2( kn +1) , kn + 1]-codes over R where k ∈ N . It is derived as the Hermitian analogue of the technique forconstructing self-dual codes first given in [8], which uses a matrix G defined by G = ( I kn +1 X ) , where X = (cid:18) x X X T Y (cid:19) , and Y = circ(( A , A , . . ., A k − )) , and where A i are circulant matrices for i ∈ [0 .. k − 1] with X = ( x , x , . . . , x ) and X =( x , x , . . . , x ) for elements x , x , x ∈ R . Theorem 3.6. Let G = ( I kn +1 X ) , where X = (cid:18) x X X T Y (cid:19) , and Y = circ(( A , A , . . ., A k − )) , and where x ∈ R , A i = circ( a i ) = circ(( a i :0 , a i :1 , . . ., a i : n − )) ∈ R n for i ∈ [0 .. k − and X =( x , x , . . . , x ) , X = ( x , x , . . . , x ) ∈ R k such that x = ( x , x , . . . , x ) , x = ( x , x , . . . , x ) ∈ R n . Then G is a generator matrix of a Hermitian self-dual [2( kn + 1) , kn + 1] -code over R if andonly if x x + knx x = 0 ,x x + x k − X i =0 n − X s =0 a i : s = 0 , k − X i =0 Θ( a i , a i , t ) = (cid:26) − − x x , t = 0 , − x x , t ∈ [1 .. ⌊ n/ ⌋ ] , k − X i =0 Θ( a [ i + j ] k , a i , 0) = − x x , j ∈ [1 .. ⌊ k/ ⌋ , k − X i =0 Θ( a [ i + j ] k , a i , t ) = − x x , j ∈ [1 .. k − ,t ∈ [1 .. ⌊ n/ ⌋ ] . Proof. We know that G is a generator matrix of a Hermitian self-dual [2( kn + 1) , kn + 1]-code over R if and only if XX T = − I kn +1 . We have XX T = (cid:18) x X X T Y (cid:19) (cid:18) x X X T Y (cid:19) T = (cid:18) x X X T Y (cid:19) (cid:18) x X X T Y T (cid:19) = (cid:18) x x + X X T x X + X Y T x X T + Y X T X T X + Y Y T (cid:19) = (cid:18) Z Z Z T Z (cid:19) so that XX T = − I kn +1 if and only if Z = − Z = and Z = − I kn . It is easy to see that Z = x x + X X T = x x + knx x , so we must have1 + x x + knx x = 0 . Let X Y T = ( u , u , . . . , u k ), where u j = ( u j :1 , u j :2 , . . . , u j : n ) ∈ R n , ∀ j ∈ [1 .. k ]. Then Z = x X + X Y T = x ( x x · · · x ) + ( u u · · · u )= ( x x + u x x + u · · · x x + u k )= ( x x + u x x + u · · · x x + u k : n )and so we see that Z = if and only if x x + u j : t = 0, ∀ j ∈ [1 .. k ] and t ∈ [1 .. n ]. Since Y =circ(( A , A , . . ., A k − )) and A i = circ(( a i :0 , a i :1 , . . ., a i : n − )), we observe that u j = x P k − i =0 A iT sothat u j : t = x P k − i =0 P n − s =0 a i : s , ∀ j ∈ [1 .. k ] and t ∈ [1 .. n ]. Thus, we find that Z = if and only if x x + x k − X i =0 n − X s =0 a i : s = 0 . Finally, since X T X = x x x x · · · x x x x x x · · · x x ... ... . . . ... x x x x · · · x x we have that Z = − I kn if and only if Y Y T = − − x x − x x · · · − x x − x x − − x x · · · − x x ... ... . . . ... − x x − x x · · · − − x x . (3.4)By following the proof of Theorem 3.4, we find that (3.4) is satisfied if and only if k − X i =0 Θ( a i , a i , t ) = (cid:26) − − x x , t = 0 , − x x , t ∈ [1 .. ⌊ n/ ⌋ ] , k − X i =0 Θ( a [ i + j ] k , a i , 0) = − x x , j ∈ [1 .. ⌊ k/ ⌋ , k − X i =0 Θ( a [ i + j ] k , a i , t ) = − x x , j ∈ [1 .. k − ,t ∈ [1 .. ⌊ n/ ⌋ ] . (cid:3) Remark . The search field for Hermitian self-dual [2( kn + 1) , kn + 1]-codes over R constructedby Theorem 3.6 is of size | R | kn +3 . 4. Results In this section, we apply the three techniques to construct best known quaternary Hermitianself-dual codes of lengths 24–40. We also apply the following well-known technique for constructingHermitian self-dual codes referred to as the building-up construction. Theorem 4.1. ([6]) Let R be a commutative Frobenius ring with a fixed non-trivial involutoryautomorphism : R → R . Let G ′ be a generator matrix of a Hermitian self-dual [2 n, n ] -code C ′ over R and let r i denote the i th row of G ′ . Let ε ∈ R : εε = − , δ ∈ R n : h δ , δ i H = − and γ i = h r i , δ i H for i ∈ [1 .. n ] . Then the matrix G = δ − γ εγ r − γ εγ r ... ... ... − γ n εγ n r n is a generator matrix of a Hermitian self-dual [2( n + 1) , n + 1] -code over R . We conduct the search for these codes using MATLAB and determine their properties using Q-extension [3]. Table 1 gives the hexadecimal notation system we use to represent elements of F and F + u F . Table 1. Hexadecimal notation system for elements of F and F + u F . F F + u F Symbol0 0 w w w w − u − u − w + u − w + u − wu − wu − w + wu A − w + wu B − u + wu C − u + wu D − w + u + wu E − w + u + wu F Using Theorems 3.1 and 3.4, we find 347 and 9 Hermitian self-dual [24 , , , , , , F constructed by Theorem 3.1, we obtain Hermitian self-dual [26 , , Using Theorem 3.1 and 3.4, we find 3 and 1 Hermitian self-dual [28 , , , , , , , , , , , , , , , , , , , , The weight enumerator of a quaternary Hermitian self-dual [26 , , W = 1 + αx + (10725 − α ) x + · · · , where α ∈ Z . The existence of codes with weight enumerator W has previously been determinedfor α ∈ { z : z = 39 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } (see [15, 22, 25]).We obtain 71 new optimal quaternary Hermitian self-dual codes of length 26 which have weightenumerator W for α ∈ { z : z = 51 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } .Of the 71 new codes, 51 are constructed by first applying Theorem 3.1 to obtain codes of length24 over F (Table 2) to which we then apply Theorem 4.1 (Table 3) and similarly 20 are constructedby first applying Theorem 3.1 to obtain codes of length 12 over F + u F (Table 4) to the image ofwhich under ϕ F + u F we then apply Theorem 4.1 (Table 5). In Table 3, we only list a sample of 25codes to save space. , , The weight enumerator of a quaternary Hermitian self-dual [32 , , W = 1 + αx + (67704 − α ) x + · · · , where α ∈ Z . The existence of codes with weight enumerator W has previously been determinedfor α ∈ { z : z = 290 , , , , , , , , , , , , , , , , , , , , , , , , , } (see [15, 22, 16]).We obtain 82 new best known quaternary Hermitian self-dual codes of length 32 which haveweight enumerator W for α ∈ { z : z = 404 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 2. Codes of length 24 over F from Theorem 3.1 to which we apply Theorem 4.1 to obtain newquaternary Hermitian self-dual [26 , , C ′ ,i λ µ a b c Table 3. Sample of new quaternary Hermitian self-dual [26 , , C ′ ,j as given in Table 2. C ,i C ′ ,j ε δ α | Aut( C ,i ) | 153 2 · 162 33 6 165 2 · 171 35 16 183 36 20 189 37 19 192 38 15 207 39 18 216 310 13 342 311 3 351 312 7 405 313 10 450 314 8 468 2 · 315 8 471 316 8 477 2 · 317 8 480 3 18 8 495 319 8 498 320 8 504 2 · 321 8 507 322 8 513 323 8 516 324 8 522 2 · 325 8 525 2 · Table 4. Codes of length 12 over F + u F from Theorem 3.1 to which we apply Theorem 4.1 to obtainnew quaternary Hermitian self-dual [26 , , C ′ ,i λ µ a b c A 1 (12A) (4EE) (B62) A 5 (A92) (AB0) (1D7) F 1 (FFA) (28F) (E95) F 5 (F4A) (B1E) (EA9) Table 5. New quaternary Hermitian self-dual [26 , , ϕ F + u F ( C ′ ,j )as given in Table 4. C ,i C ′ ,j ε δ α | Aut( C ,i ) | 26 3 174 327 7 180 328 1 198 329 15 201 330 8 219 331 14 225 332 4 261 333 6 363 334 5 369 335 13 387 336 2 396 337 10 423 338 12 459 339 9 594 340 11 630 2 · 41 10 639 2 · 342 11 660 2 · 343 10 666 2 · 44 10 795 2 · 45 10 840 2 · , , , , , , , , , , , , , , , , , , , , , , } .Of the 82 new codes, 70 are constructed by applying Theorem 3.1 over F (Table 6) and 12 areconstructed by applying Theorem 3.1 over F + u F (Table 7). In Table 6, we only list a sample of25 codes to save space. , , The weight enumerator of a quaternary Hermitian self-dual [36 , , W = 1 + αx + (771120 − α ) x + · · · , Table 6. Sample of new quaternary Hermitian self-dual [32 , , F . C ,i λ µ a b c α | Aut( C ,i ) | · · · · · · · Table 7. New quaternary Hermitian self-dual [32 , , F + u F . C ,i λ µ a b c α | Aut( C ,i ) | · · F 3 (34CB) (38A9) (8A80) · F 1 (ECD5) (1AC4) (3C0C) · · F F (3368) (DEEF) (A31C) · A 3 (C6BD) (1A26) (09AB) · · · · F 1 (ADE2) (E9C3) (F0FF) · · α ∈ Z . The existence of codes with weight enumerator W has previously been determinedfor α ∈ { z : z = 2316 , } (see [15, 16]).We obtain 2 new best known quaternary Hermitian self-dual codes of length 36 which have weightenumerator W for α ∈ { z : z = 2172 , } .The new codes are constructed by applying Theorem 3.1 over F (Table 8). Table 8. New quaternary Hermitian self-dual [36 , , F . C ,i λ µ a b c α | Aut( C ,i ) | · · , , The weight enumerator of a quaternary Hermitian self-dual [38 , , W = 1 + αx + (430236 − α ) x + · · · , where α ∈ Z . The existence of codes with weight enumerator W has previously been determinedfor α ∈ { z : z = 3249 , , , , , , , , , , , , , } (see [15, 16]).We obtain 1 new best known quaternary Hermitian self-dual code of length 38 which has weightenumerator W for α ∈ { z : z = 3384 } .The new code is constructed by applying Theorem 3.6 over F (Table 9). Table 9. New quaternary Hermitian self-dual [38 , , F . C ,i k x x x a a a α | Aut( C ,i ) | · , , The weight enumerator of a quaternary Hermitian self-dual [40 , , W = 1 + αx + (232560 − α ) x + · · · , where α ∈ Z . The existence of codes with weight enumerator W has previously been determinedfor α ∈ { z : z = 1387 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } (see [15, 16]).We obtain 252 new best known quaternary Hermitian self-dual codes of length 40 which haveweight enumerator W for α ∈ { z : z = 1265 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,..., , , , , , , ..., , , , , , , , , , , , , , , , , , , , , ..., , , , , , , , , , , , , ..., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } . Of the 252 new codes, 182 are constructed by applying Theorem 3.1 over F (Table 10); 67 areconstructed by applying Theorem 3.1 over F + u F (Table 11) and 3 are constructed by applyingTheorem 3.4 over F (Table 12). In both Tables 10 and 11, we only list a sample of 25 codes to savespace. Table 10. Sample of new quaternary Hermitian self-dual [40 , , F . C ,i λ µ a b c α | Aut( C ,i ) | · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Conclusion In this work, we presented three techniques for constructing Hermitian self-dual codes using λ -circulant matrices. The first technique was derived as the Hermitian analogue of a modified fourcirculant technique for constructing self-dual codes given in [14]. The other two techniques werederived as the Hermitian analogues of the techniques for constructing self-dual codes given in [8].We proved the necessary conditions for these techniques to produce Hermitian self-dual codes usinga specialised mapping Θ which is inherently associated with λ -circulant matrices. We explored theability of these techniques by implementing them to construct best known and optimal quaternaryHermitian self-dual codes which were previously not known to exist. In particular, we were able toconstruct new optimal codes of length 26 and many best known codes of lengths 32, 36, 38 and 40.The performance of the first technique in searching for Hermitian self-dual codes was far superiorto that of the other techniques. However, the first technique is restricted in that it can only be usedto construct codes of length 4 n . The main advantage of using the other techniques is being able toconstruct codes of a greater variety of lengths. On the other hand, the efficiency of the second andthird techniques drops drastically as length increases.We were unable to improve on the best known minimum distance of quaternary Hermitian self-dual codes for lengths 32–40. Due to computational limitations, we did not investigate constructingcodes of lengths greater than 40. Moreover, the codes we did construct were obtained by randomsearches alone. However, as in [14], the implementation of the mapping Θ enabled us to reduce the Table 11. Sample of new quaternary Hermitian self-dual [40 , , F + u F . C ,i λ µ a b c α | Aut( C ,i ) | A 3 (11C05) (E31C7) (7D9F6) · F F (69F28) (66279) (01084) · · A 3 (A4E4F) (42F5E) (B5B25) · · · · · A A (DF57B) (60252) (1B31E) · A 2 (EF44A) (68E3B) (04140) · F 2 (4ED9F) (84931) (ABD1F) · · · · · · F 2 (F7ABF) (4C89B) (B1AD5) · · · · F 3 (09409) (2F629) (7E37E) · A 2 (8FF41) (3FC7D) (9179E) · F 2 (6C1A6) (BFC90) (96D1E) · · · Table 12. New quaternary Hermitian self-dual [40 , , F . 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