Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes
aa r X i v : . [ c s . I T ] J a n Construction of binary LCD codes, ternaryLCD codes and quaternary Hermitian LCDcodes
Masaaki Harada ∗ January 29, 2021
Abstract
We give two methods for constructing many linear complementarydual (LCD for short) codes from a given LCD code, by modifying someknown methods for constructing self-dual codes. Using the methods,we construct binary LCD codes and quaternary Hermitian LCD codes,which improve the previously known lower bound on the largest min-imum weights.
Let F q denote the finite field of order q , where q is a prime power. An [ n, k ]code C over F q is called (Euclidean) linear complementary dual if C ∩ C ⊥ = { n } , where C ⊥ is the dual code of C and n denotes the zero vector oflength n . An [ n, k ] code C over F q is called Hermitian linear complementarydual if C ∩ C ⊥ H = { n } , where C ⊥ H is the Hermitian dual code of C . Thesetwo families of codes are collectively called linear complementary dual (LCDfor short) codes.LCD codes were introduced by Massey [22] and gave an optimum linearcoding solution for the two user binary adder channel. In recent years, there ∗ Research Center for Pure and Applied Mathematics, Graduate School of InformationSciences, Tohoku University, Sendai 980–8579, Japan. email: [email protected] . F q is equivalent to some LCD code for q ≥ F q is equivalent to some Hermitian LCD code for q ≥
3. Thismotivates us to study binary LCD codes, ternary LCD codes and quaternaryHermitian LCD codes.Meanwhile, self-dual codes are one of the most interesting classes of codes.For example, this interest is justified by many combinatorial objects and al-gebraic objects related to self-dual codes (see e.g. [23] for basic facts con-cerning self-dual codes). Many methods for constructing self-dual codes areknown. Although the definitions say that self-dual codes and LCD codes arequite different classes of codes, codes of both classes are characterized bytheir generator matrices. In this situation, it is natural to consider methodsfor constructing LCD codes by modifying some known methods of self-dualcodes. Also, it is a fundamental problem to determine the largest minimumweight among all [ n, k ] codes in a certain class of codes for a given pair ( n, k ).In this paper, we give two methods for constructing many LCD codes froma given LCD code by modifying known methods of constructing many self-dual codes from a given self-dual code. Using the methods, we constructbinary LCD codes and quaternary Hermitian LCD codes, which improve thepreviously known lower bound on the largest minimum weights.The paper is organized as follows. In Section 2, we give some definitions,notations and basic results used in this paper. In Section 3, we give a methodfor constructing binary LCD [ n + 2 , k + 1] codes, ternary LCD [ n + 3 , k + 1]codes and quaternary Hermitian LCD [ n + 2 , k + 1] codes from a given binaryLCD [ n, k ] code, ternary LCD [ n, k ] code and quaternary Hermitian LCD[ n, k ] code, respectively. This is established by modifying the method of self-dual codes in [13]. In addition, we give a method for constructing binaryeven LCD [ n, k ] codes from a given binary even LCD [ n, k ] code. This isestablished by modifying the methods of self-dual codes in [12] and [15]. InSection 4, we construct binary LCD codes with parameters:[28 , , , [28 , , , [29 , , , [30 , , , [30 , , , [32 , , , [32 , , , [33 , , , [34 , , , [34 , , , [34 , , , [36 , , , [36 , , , [40 , ,
18] and [40 , , . d ( n, k ) = n, k ) = (32 , , (32 , , (33 , , (34 , , (34 ,
22) and (36 , , n, k ) = (28 , , (29 ,
13) and (30 , ,
12 if ( n, k ) = (28 , , (30 ,
7) and (34 , ,
16 if ( n, k ) = (36 ,
6) and (40 , ,
18 if ( n, k ) = (40 , , where d ( n, k ) denotes the largest minimum weight among all binary LCD[ n, k ] codes. In Section 5, we construct quaternary Hermitian LCD codeswith parameters:[21 , , , [21 , , , [21 , , , [22 , , , [23 , , , [24 , , , [25 , , , [25 , , , [26 , , , [26 , , , [26 , , , [26 , , , [27 , , , [27 , , , [28 , , , [28 , , , [28 , , , [29 , , , [29 , , , [30 , ,
4] and [30 , , . These codes improve the previously known lower bounds on the largest min-imum weights. In particular, we have d H ( n, k ) = ( n, k ) = (23 , , (26 , , (27 , , (28 ,
23) and (30 , , n, k ) = (24 , , where d H ( n, k ) denotes the largest minimum weight among all quaternaryHermitian LCD [ n, k ] codes. Also, the above quaternary Hermitian LCDcodes show the new existence of some entanglement-assisted quantum codes.In Section 6, we give examples of ternary LCD codes constructed by the firstmethod in Section 3, which have minimum weights meeting the lower boundson the largest minimum weights among currently known all ternary codes. Let F q denote the finite field of order q , where q is a prime power. An [ n, k ]code C over F q is a k -dimensional vector subspace of F nq . The parameter n is called the length of C . A matrix whose rows are linearly independent and3enerate C is called a generator matrix of C . The weight wt( x ) of a vector x of F nq is the number of non-zero coordinates in x . The minimum weight of C is defined as min { wt( x ) | n = x ∈ C } , where n denotes the zero vectorof length n . An [ n, k, d ] code over F q is an [ n, k ] code over F q with minimumweight d . The elements of C are called codewords . Two codes C and C ′ over F q are equivalent if there is a monomial matrix P over F q with C ′ = C · P ,where C · P = { xP | x ∈ C } .The dual code C ⊥ of an [ n, k ] code C over F q is defined as C ⊥ = { x ∈ F nq |h x, y i = 0 for all y ∈ C } , where h x, y i = P ni =1 x i y i for x = ( x , x , . . . , x n ) , y =( y , y , . . . , y n ) ∈ F nq . For any element α of F q , the conjugation of α is de-fined as α = α q . The Hermitian dual code C ⊥ H of an [ n, k ] code C over F q is defined as C ⊥ H = { x ∈ F nq | h x, y i H = 0 for all y ∈ C } , where h x, y i H = P ni =1 x i y i for x = ( x , x , . . . , x n ) , y = ( y , y , . . . , y n ) ∈ F nq . A code C over F q is called (Euclidean) linear complementary dual if C ∩ C ⊥ = { n } . A code C over F q is called Hermitian linear complementary dual if C ∩ C ⊥ H = { n } .These two families of codes are collectively called linear complementary dual (LCD for short) codes.Throughout this paper, we employ the following notations of vectors andmatrices. Let I k denote the identity matrix of order k . Let k denote theall-one vector of length k . Let A T denote the transpose of a matrix A . Fora matrix A = ( a ij ), the conjugate matrix of A is defined as A = ( a ij ).The following characterization is due to Massey [22] (see e.g. [11] forHermitian LCD codes). Lemma 2.1. (i)
Let C be a code over F q . Let G be a generator matrix of C . Then C is LCD if and only if GG T is nonsingular. (ii) Let C be a code over F q . Let G be a generator matrix of C . Then C is Hermitian LCD if and only if GG T is nonsingular. Codes over F , F and F are called binary, ternary and quaternary , re-spectively. A binary code C is called even if the weights of all codewords of C are even. In this paper, we denote the finite fields F and F by { , } and { , , } , respectively, and we denote the finite field F by { , , ω, ω } , where ω = ω + 1.Throughout this paper, let d q ( n, k ) denote the largest minimum weightamong all LCD [ n, k ] codes over F q ( q = 2 , d H ( n, k ) denote thelargest minimum weight among all quaternary Hermitian LCD [ n, k ] codes.4he following bound was proved in [6, Theorem 8] for binary LCD codes andin [17, Theorem 1] for ternary LCD codes and quaternary Hermitian LCDcodes. We remark that the constructive proofs are given in [6] and [17]. Lemma 2.2.
Suppose that ≤ k ≤ n . Then d q ( n, k ) ≤ d q ( n, k − for q ∈ { , } and d H ( n, k ) ≤ d H ( n, k − . We end this section by giving self-dual codes for the comparison withLCD codes. A code C over F q is called (Euclidean) self-dual if C = C ⊥ . Acode C over F q is called Hermitian self-dual if C = C ⊥ H . These two familiesof codes are collectively called self-dual codes. Self-dual codes are one ofthe most interesting classes of codes (see e.g. [23] for basic facts concerningself-dual codes). It is known that a [2 n, n ] code C over F q (resp. F q ) is self-dual (resp. Hermitian self-dual) if and only if GG T = O n (resp. GG T = O n )for a generator matrix G of C , where O n is the n × n zero matrix. Weemphasis that both LCD codes and self-dual codes with generator matrices G are characterized by GG T or GG T . In this section, we give two methods for constructing many LCD codes froma given LCD code.
Starting from a given self-dual [2 n, n ] code, some methods for constructingmany self-dual [2 n + 2 , n + 1] codes are known (see e.g. [13] and [18]). Bymodifying the method in [13], we give the following method. Theorem 3.1. (i)
Let C be a binary LCD [ n, k ] code with generator ma-trix G . Let x = ( x , x , . . . , x n ) be a vector of F n . Let C ( x ) be thebinary code with the following generator matrix: G ( x ) = x · · · x n h x, r i h x, r i ... ... G h x, r k i h x, r k i , here r i is the i -th row of G . If wt( x ) is even, then C ( x ) is a binaryLCD [ n + 2 , k + 1] code. (ii) Let C be a ternary LCD [ n, k ] code with generator matrix G . Let x = ( x , x , . . . , x n ) be a vector of F n , and let a = ( a , a , a ) be a vectorof F . Let C ( x, a ) be the ternary code with the following generatormatrix: G ( x, a ) = a a a x · · · x n h x, r i h x, r i h x, r i ... ... ... G h x, r k i h x, r k i h x, r k i , where r i is the i -th row of G . If either wt( x ) and a =(1 , , or wt( x ) and a = (1 , , , then C ( x, a ) is aternary LCD [ n + 3 , k + 1] code. (iii) Let C be a quaternary Hermitian LCD [ n, k ] code with generator ma-trix G . Let x = ( x , x , . . . , x n ) be a vector of F n . Let C ( x ) be thequaternary code with the following generator matrix: G ( x ) = x · · · x n h x, r i H h x, r i H ... ... G h x, r k i H h x, r k i H , where r i is the i -th row of G . If wt( x ) is even, then C ( x ) is a qua-ternary Hermitian LCD [ n + 2 , k + 1] code.Proof. All cases are similar, and we give details only for (iii).Let r ′ i denote the i -th row of G ( x ). From the construction of G ( x ), itfollows that h r ′ , r ′ i H = 1 , h r ′ , r ′ i i H = h x, r i − i H + h x, r i − i H = 0 ( i = 2 , , . . . , k + 1) , h r ′ i , r ′ i i H = h r i − , r i − i H ( i = 2 , , . . . , k + 1) , h r ′ i , r ′ j i H = h r i − , r j − i H ( i, j = 2 , , . . . , k + 1 , i = j ) . G ( x ) G ( x ) T = · · · G G T , which completes the proof of (iii). Remark . When G q ( q = 2 , ,
4) has the form (cid:0) I k A (cid:1) , we may assumewithout loss of generality that x = · · · = x k = 0. This substantially reducesthe number of the possible vectors x .In Sections 4 and 5, we construct binary LCD codes and quaternary Her-mitian LCD codes, respectively, which improve the previously known lowerbounds on the largest minimum weights by the above method. Theorem 3.1seems to be less useful for ternary LCD codes. In Section 6, we give ternaryLCD codes, which have minimum weights meeting the lower bounds on thelargest minimum weights among currently known all ternary codes. Now starting from a binary self-dual [2 n, n ] code with generator matrix ofthe form (cid:0) I n A (cid:1) , by transforming A , some methods for constructing manyself-dual [2 n, n ] codes are known (see e.g. [12], [15] and [24]). By modifyingthe methods in [12] and [15], we give a method for constructing many binaryeven LCD codes from a given binary even LCD code. Theorem 3.3.
Let C be a binary even LCD [ n, k ] code with generator matrix (cid:0) I k A (cid:1) . Let r i be the i -th row of A . Let x be a vector of F n − k . Define a k × n − k matrix A ( x ) , where the i -th row r ′ i is defined as follows: r ′ i = r i + x + h r i , x i n − k . Let C ( A, x ) be the binary [ n, k ] code with the following generator matrix: (cid:0) I k A ( x ) (cid:1) . If both wt( x ) and n − k are even, then C ( A, x ) is a binary even LCD [ n, k ] code. roof. Since wt( x ) is even, we have h x, x i = 0 and h x, n − k i = 0. Since n − k is even, we have h n − k , n − k i = 0. Since C is even, we have h r i , n − k i = 1.Hence, we have h r ′ i , r ′ j i = h r i + x + h r i , x i n − k , r j + x + h r j , x i n − k i = h r i , r j i + h r i , x i + h r j , x i + h x, r j i + h r i , x i = h r i , r j i . This shows that (cid:0) I k A (cid:1) (cid:0) I k A (cid:1) T = (cid:0) I k A ( x ) (cid:1) (cid:0) I k A ( x ) (cid:1) T . Theresult follows. Remark . If there is a binary even LCD [ n, k ] code, then k must be even [6,Theorem 5].The above method constructs 2 n − k − binary even LCD [ n, k ] codes froma given binary even LCD [ n, k ] code. It may be possible for the minimumweight of a binary even LCD [ n, k ] code to be larger than that of a givenbinary even LCD [ n, k ] code. In this case, the minimum weight is increasedby at least 2. In Section 4, we give such examples of binary even LCD codes,which improves the previously known lower bounds on the largest minimumweights. This illustrates the effectiveness of Theorem 3.3. A classification of binary LCD codes was done in [1] for n ∈ { , , . . . , } .The largest minimum weights d ( n, k ) were determined for n ∈ { , , . . . , } (see [9, Table 1] for n ∈ { , , . . . , } , [16, Table 3] for n ∈ { , , , } and [2, Table 15] for n ∈ { , , . . . , } ). For n ∈ { , , . . . , } , thecurrent information on d ( n, k ) can be found in [8, Table 2]. In this section,by Theorems 3.1 and 3.3, we construct binary LCD codes, which improve thepreviously known lower bounds on the largest minimum weights d ( n, k ). Allcomputer calculations in this section were done with the help of Magma [5].8 .1 Lengths n with ≤ n ≤ By the
Magma function
BestKnownLinearCode , one can construct a binary[27 , ,
8] code C , . The code C , has the following generator matrix: G , = I ... ... A ′ , , where A ′ , is listed in Figure 1. Using Lemma 2.1, we verified by Magma that C , is LCD. Let C , be the binary [28 ,
13] with the following generatormatrix: G , = · · · · · ·
10 1 0... I ... ... A ′ , . Since each row of A ′ , has even weight, we have G , G T , = · · · G , G T , . Thus, by Lemma 2.1, C , is LCD. In addition, we verified by Magma that C , has minimum weight 8. Lemma 4.1.
There is a binary LCD [28 , , code. By applying Theorem 3.1 to the generator matrices G , and G , of C , and C , , it is possible to construct 2 binary LCD [29 ,
13] codes andbinary LCD [30 ,
14] codes, respectively. In particular, our computer searchby
Magma found a binary LCD [29 , ,
8] code C , as C , ( x ) and a binaryLCD [30 , ,
8] code C , as C , ( x ′ ), where x = (0 , . . . , , , , , , , , , , , , , , , ,
0) and x ′ = (0 , . . . , , , , , , , , , , , , , , , , , . Lemma 4.2.
There is a binary LCD [29 , , code. There is a binary LCD [30 , , code. ′ , = A ′ , = Figure 1: Matrices A ′ , and A ′ , By using the method in [16, Section 6], our computer search by
Magma found a binary even LCD [28 , ,
10] code C ′ , with generator matrix G ′ , = (cid:0) I A ′ , (cid:1) , where A ′ , is listed in Figure 1. By applying Theorem 3.3to the matrix A ′ , , it is possible to construct 2 binary even LCD [28 , Magma found a binary evenLCD [28 , ,
12] code D , as C ′ , ( A ′ , , x ), where x = (1 , , , , , , , , . . . , . Note that the minimum weight of D , is increased by 2. Lemma 4.3.
There is a binary even LCD [28 , , code. Again, by applying Theorem 3.1 to the generator matrix G ′ , of D , ,our computer search by Magma found a binary even LCD [30 , ,
12] code C ′ , as D , ( x ), where x = (0 , . . . , , , , , , , , , , , , , , , , , , , , , , , . Lemma 4.4.
There is a binary LCD [30 , , code. .2 Lengths n with ≤ n ≤ Now let us look at construction of binary LCD [ n, k ] codes for 31 ≤ n ≤ d ( n, k ) is known for n ≥ Lemma 4.5.
For ( n, k, d ) ∈ P , where P = { (34 , , , (36 , , , (40 , , , (40 , , } , (1) there is a binary even LCD [ n, k, d ] code. For ( n, k, d ) ∈ P , we describe how the above binary even LCD [ n, k, d ]codes C , , C , , C , and C ′ , were constructed, respectively. As the firststep, by using the method in [16, Section 6], our computer search by Magma found a binary even LCD [ n, k, d −
2] code with generator matrix (cid:0) I k A (cid:1) for ( n, k, d ) ∈ P in (1). Our feeling is that construction of a binary evenLCD [ n, k, d −
2] code is usually easier than that of a binary even LCD [ n, k, d ]code by the method. As the next step, by applying Theorem 3.3 to A , ourcomputer search by Magma found a binary even LCD [ n, k, d ] code. Notethat the minimum weight is increased by 2 in this case. The codes C , , C , , C , and C ′ , have the following generator matrices: (cid:0) I A , (cid:1) , (cid:0) I A , (cid:1) , (cid:0) I A , (cid:1) and (cid:0) I A ′ , (cid:1) , respectively, where A , , A , , A , and A ′ , are listed in Figure 5.Now, by the Magma function
BestKnownLinearCode , one can constructa binary [32 , ,
6] code D , and a binary [34 , ,
6] code D , . The codes D , and D , have the following generator matrices: G , = (cid:0) I B , (cid:1) and G , = (cid:0) I B , (cid:1) , respectively, where B , and B , are listed in Figure 2. Using Lemma 2.1,we verified by Magma that D , and D , are LCD. By applying Theo-rem 3.1 to G , and G , , our computer search by Magma found a binaryLCD [34 , ,
6] code D ′ , as D , ( x ) and a binary LCD [36 , ,
6] code D , as D , ( x ′ ), where x = (0 , . . . , , , , , , , , , , , , ,
0) and x ′ = (0 , . . . , , , , , , , , , , , , . Lemma 4.6.
For ( n, k ) ∈ { (32 , , (34 , , (34 , , (36 , } , there is abinary LCD [ n, k, code. , = B , = Figure 2: Matrices B , and B , emma 4.7. There is a binary LCD [33 , , code.Proof. The second column of B , is T . Let D , be the binary [33 , , D , as the punctured code by deleting the 24-thcoordinate. It is obvious that D , is LCD.Let ˆ d ( n, k ) denote the largest minimum weight among all binary [ n, k ]codes. The current information on ˆ d ( n, k ) can be found in [10]. For example,it is known thatˆ d (34 ,
10) = 12 , ˆ d (36 ,
6) = ˆ d (40 ,
8) = 16 and ˆ d (40 ,
6) = 18 . (2) Lemma 4.8.
There is a binary LCD [32 , , code.Proof. By Lemma 4.6, there is a binary LCD [32 , ,
6] code. It follows fromLemma 2.2 with (2) that6 ≤ d (32 , ≤ d (32 , ≤ ˆ d (32 ,
19) = 6 . The result follows. d ( n, k ) It is previously known [8] that d ( n, k ) ∈ ( { , } if ( n, k ) = (28 , ,
13) and (30 , { , } if ( n, k ) = (28 ,
6) and (30 , d ( n, k ) = 6 if ( n, k ) = (32 , , (32 , , (33 , , (34 , , (34 ,
22) and (36 , . (4)From Lemmas 4.1–4.8 with (2)–(4), we have the following: Theorem 4.9.
Let d ( n, k ) denote the largest minimum weight among allbinary LCD [ n, k ] codes. Then d ( n, k ) = if ( n, k ) = (32 , , (32 , , (33 , , (34 , , (34 , and (36 , , if ( n, k ) = (28 , , (29 , and (30 , , if ( n, k ) = (28 , , (30 , and (34 , , if ( n, k ) = (36 , and (40 , , if ( n, k ) = (40 , .
13f there is a binary LCD [ n, k, d ] code with k ≥
3, then there is a bi-nary LCD [ n + (2 k − s, k, d + 2 k − s ] code for every positive integer s [3,Lemma 3.5]. Hence, as a consequence of Lemmas 4.1–4.8, we have the fol-lowing: Corollary 4.10.
Suppose that ( n, k, d ) ∈ P ′ , where P ′ = (28 , , , (28 , , , (29 , , , (30 , , , (30 , , , (32 , , , (32 , , , (33 , , , (34 , , , (34 , , , (34 , , , (36 , , , (36 , , , (40 , , , (40 , , . For a nonnegative integer s , there is a binary LCD [ n + (2 k − s, k, d + 2 k − s ] code. The current information on the largest minimum weights d H ( n, k ) can befound in [21] for n ≤
25 (see also [14]). In this section, by Theorem 3.1we construct quaternary Hermitian LCD codes, which improve the previ-ously known lower bounds on the largest minimum weights d H ( n, k ). Thesenew quaternary Hermitian LCD codes establish the new existence of someentanglement-assisted quantum codes. All computer calculations in this sec-tion were done with the help of Magma [5]. n = 21 and There are quaternary Hermitian LCD codes with parameters [19 , ,
9] and[20 , ,
10] [14]. We denote these codes by C , and C , , respectively. Thecode C , has generator matrix G , = (cid:0) I M (cid:1) , where M is listedin [14, Fig. 1]. Since C , is constructed from C , as the punctured code bydeleting the first coordinate [14], C , has the following generator matrix: G , = (cid:18) · · · I M (cid:19) . By applying Theorem 3.1 to the generator matrices G , and G , of thecodes C , and C , , it is possible to construct quaternary Hermitian LCD[21 ,
8] codes and quaternary Hermitian LCD [22 ,
8] codes, respectively. In14articular, our computer search by
Magma found a quaternary Hermi-tian LCD [21 , ,
9] code C , as C , ( x ) and a quaternary Hermitian LCD[22 , ,
10] code C , as C , ( x ′ ), where x = (0 , . . . , , ω, ω, , , ω, , ω, ω, ω, , ω ) and x ′ = (0 , . . . , , ω, , , ω, , ω, , , , ω, , ω, ω ) . Lemma 5.1.
There is a quaternary Hermitian LCD [22 , , code. Thereis a quaternary Hermitian LCD [21 , , code. By the
Magma function
BestKnownLinearCode , one can construct aquaternary [19 , ,
8] code D , . Using Lemma 2.1, we verified by Magma that D , is Hermitian LCD. The code D , has generator matrix G , = (cid:0) I A , (cid:1) , where A , is listed in Figure 3. By using the method in [14,Section 2], our computer search by Magma found a quaternary HermitianLCD [19 , ,
7] code D ′ , with generator matrix G ′ , = (cid:0) I A ′ , (cid:1) , where A ′ , is listed in Figure 3. By applying Theorem 3.1 to the generator matri-ces G , and G ′ , of the codes D , and D ′ , , it is possible to constructquaternary Hermitian LCD [21 ,
10] codes and quaternary Hermitian LCD[21 ,
11] codes, respectively. In particular, our computer search by
Magma found a quaternary Hermitian LCD [21 , ,
8] code D , as D , ( x ) and aquaternary Hermitian LCD [21 , ,
7] code D ′ , as D ′ , ( x ′ ), where x = (0 , . . . , , , ω, , ω, ω, ω, ω, , ω,
1) and x ′ = (0 , . . . , , ω, , ω, , , ω, ω, , . Lemma 5.2.
There is a quaternary Hermitian LCD [21 , , code. Thereis a quaternary Hermitian LCD [21 , , code. n with ≤ n ≤ Now let us look at construction of quaternary Hermitian LCD [ n, k ] codesfor 23 ≤ n ≤
30. Note that no information on d H ( n, k ) is known for n ≥ n, k, d ) ∈ P , where P = { (23 , , , (24 , , , (26 , , , (27 , , , (28 , , } . (5)By the Magma function
BestKnownLinearCode , one can construct a quater-nary [ n, k, d ] code. By considering an equivalent code, we have a quaternary15 , = ω ω ω ω ω ωω ω ω ω ω ω ω ωω ω ω ω ω ωω ω ω ω ω ω ω ω ω ωω ω ω ω ω ωω ω ω ω ω ω ω ωω ω ω ω ω ω A ′ , = ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω Figure 3: Matrices A , and A ′ , [ n, k, d ] code C ,n with the following generator matrix: G ,n = (cid:0) I k A ,n (cid:1) , where A ,n is listed in Figure 4. Using Lemma 2.1, we verified by Magma that C ,n is Hermitian LCD. By applying Theorem 3.1 to G ,n , our computersearch by Magma found a quaternary Hermitian LCD [ n + 2 , k + 1 , d ] code C ′ ,n +2 as C ,n ( x n ), where x = (0 , . . . , , ω, , ω, ω, ω ) , x = (0 , . . . , , , ω, ω, ω, ω, ω ) ,x = (0 , . . . , , ω, , , , , x = (0 , . . . , , ω, ω, , ω,
1) and x = (0 , . . . , , ω, , , ω, . Lemma 5.3.
For ( n, k, d ) ∈ P in (5) , there is a quaternary Hermitian LCD [ n, k, d ] code and there is a quaternary Hermitian LCD [ n + 2 , k + 1 , d ] code. , = ω ωω ω ωω ω ω ω ω ω ω ω ω ω ωω ω ωω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ωω ω ω ω ω ω ω ω ω ω ω ω A , = ω ω ω ω ωω ω ω ω ω ω ωω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ωω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ωω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω A , = ω ω ω ω ω ω ω ω ωω ω ωω ω ω ω ωω ω ω ω ω ω ω ω ω ωω ω ω ω ω ω ω ω ω ω ω ω ω ω ωω ω ω ω
00 1 ω ω
11 1 0 1 ω ω ω ωω ω ω ω ωω ω ω A , = ω ω ω ω ω ω ω ωω ω ω ω ω ω ωω ω ω ωω ω ω ω ω ω ω ω ω ω ω
01 0 0 ω ωω ω ω ω ω ω ω ω ω ωω ω ω ω ω ω
11 0 0 1 10 ω ω ω ωω ω ω ωω ω ω A , = ω ω ω ω ω ω ω ω ω ω ω ω ω ωω ω ω ωω ω ω ω ω ω ω ω ω ω ω ω ω ωω ω ωω ω ωω ω ω ω ω ωω ω ω ω ω ωω ω ω ω
01 0 ω ω ω ω ω ω ω ω ω Figure 4: Matrices A , , A , , A , , A , and A , .3 Determination of d H ( n, k ) and bounds on d H ( n, k ) It is previously known [21] that d H (21 , ∈ { , , } , d H (21 , ∈ { , , } ,d H (21 , ∈ { , , } and d H (22 , ∈ { , , } . (6)Let ˆ d ( n, k ) denote the largest minimum weight among all quaternary [ n, k ]codes. The current information on ˆ d ( n, k ) can be found in [10]. For example,it is known thatˆ d ( n, k ) = n, k ) = (23 , , (26 , , (27 , , (28 ,
23) and (30 , , n, k ) = (24 , , andˆ d ( n, k ) ∈ ( { , } if ( n, k ) = (25 , , (28 ,
22) and (29 , , { , } if ( n, k ) = (26 , . (7)From Lemmas 5.1–5.3 with (6) and (7), we have the following: Theorem 5.4.
Let d H ( n, k ) denote the largest minimum weight among allquaternary Hermitian LCD [ n, k ] codes. Then d H ( n, k ) = ( if ( n, k ) = (23 , , (26 , , (27 , , (28 , and (30 , , if ( n, k ) = (24 , , and d H ( n, k ) ∈ { , } if ( n, k ) = (25 , , (28 , and (29 , , { , } if ( n, k ) = (26 , , { , } if ( n, k ) = (21 , , { , } if ( n, k ) = (21 , , { , } if ( n, k ) = (21 , , { , } if ( n, k ) = (22 , . Corollary 5.5. d H ( n, k ) ∈ { , } if ( n, k ) = (26 , and (27 , , { , , } if ( n, k ) = (25 , , (28 , , (29 , and (30 , , { , , } if ( n, k ) = (26 , . roof. It is known [10] that ˆ d ( n, k ) ≤ n, k ) = (26 ,
20) and (27 , d ( n, k ) ≤ n, k ) = (25 , , ,
22) and (30 , d (26 , ≤
8. The result follows from Lemma 2.2 and Theorem 5.4.If there is a quaternary Hermitian LCD [ n, k, d ] code with k ≥
2, thenthere is a quaternary Hermitian LCD [ n + k − s, k, d + 4 k − s ] code for everypositive integer s [2, Lemma 3.3]. Hence, we have the following: Corollary 5.6.
Suppose that ( n, k, d ) ∈ P ′ , where P ′ = (21 , , , (21 , , , (21 , , , (22 , , , (23 , , , (24 , , , (25 , , , (25 , , , (26 , , , (26 , , , (26 , , , (26 , , , (27 , , , (27 , , , (28 , , , (28 , , , (28 , , , (29 , , , (29 , , , (30 , , , (30 , , . (8) For a nonnegative integer s , there is a quaternary Hermitian LCD [ n + k − s, k, d + 4 k − s ] code. An entanglement-assisted quantum [[ n, k, d ; c ]] code C encodes k informationqubits into n channel qubits with the help of c pairs of maximally entan-gled Bell states. The parameter d is called the minimum weight of C . Theentanglement-assisted quantum code C can correct up to ⌊ d − ⌋ errors actingon the n channel qubits (see e.g. [19] and [20]). An entanglement-assistedquantum [[ n, k, d ; 0]] code is a standard quantum code. If there is a qua-ternary Hermitian LCD [ n, k, d ] code, then there is an entanglement-assistedquantum [[ n, k, d ; n − k ]] code (see e.g. [19] and [20]). Hence, as a consequenceof Corollary 5.6, we have the following: Corollary 5.7.
Suppose that ( n, k, d ) ∈ P ′ , where P ′ is listed in (8) . Fora nonnegative integer s , there is an entanglement-assisted quantum [[ n + k − s, k, d + 4 k − s ; n + k − s − k ]] code. A classification of ternary LCD codes was done in [1] for n ∈ { , , . . . , } .The largest minimum weights d ( n, k ) were determined for n ∈ { , , . . . , } Magma [5].Let ˆ d ( n, k ) denote the largest minimum weight among all ternary [ n, k ]codes. The current information on ˆ d ( n, k ) can be found in [10]. For example,it is known that ˆ d (34 , ∈ { , } , ˆ d (37 , ∈ { , , } , ˆ d (37 ,
29) = 5 and ˆ d (40 , ∈ { , } . (9)By the Magma function
BestKnownLinearCode , one can construct a ternary[34 , ,
7] code C , and a ternary [37 , ,
5] code C , . The codes C , and C , have the following generator matrices: G , = (cid:0) I A , (cid:1) and G , = (cid:0) I A , (cid:1) , respectively, where A , and A , are listed in Figure 6. Using Lemma 2.1,we verified by Magma that C , and C , are LCD. By applying Theo-rem 3.1 to G , and G , , our computer search by Magma found a ternaryLCD [37 , ,
7] code C ′ , as C , ( x, (1 , , , , C , as C , ( x ′ , (1 , , x = (0 , . . . , , , , , , , , , , , ,
0) and x ′ = (0 , . . . , , , , , , , , , . Lemma 6.1.
For ( n, k, d ) ∈ P , where P = { (34 , , , (37 , , , (37 , , , (40 , , } , (10) there is a ternary LCD [ n, k, d ] code. From Lemma 6.1 with (9), we have the following:
Proposition 6.2.
Let d ( n, k ) denote the largest minimum weight among allternary LCD [ n, k ] codes. Then d (34 , ∈ { , } , d (37 , ∈ { , , } ,d (37 ,
29) = 5 and d (40 , ∈ { , } . orollary 6.3. d (34 , ∈ { , , } , d (37 , ∈ { , , } ,d (37 , ∈ { , } and d (40 , ∈ { , , } . Proof.
It is known [10] that ˆ d (34 , ≤
9, ˆ d (37 , ≤
9, ˆ d (37 , ≤ d (40 , ≤
7. The result follows from Lemma 2.2 and Proposition 6.2.If there is a ternary LCD [ n, k, d ] code with k ≥
2, then there is a ternaryLCD [ n + k − s, k, d + 3 k − s ] code for every positive integer s [3, Lemma 3.5].Hence, we have the following: Corollary 6.4.
Suppose that ( n, k, d ) ∈ P ∪ P ′ , where P is listed in (10) and P ′ = { (34 , , , (37 , , , (37 , , , (40 , , } . For a nonnegative integer s , there is a ternary LCD [ n + k − s, k, d + 3 k − s ] code. Acknowledgments.
This work was supported by JSPS KAKENHI GrantNumber 19H01802.
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