Abstract
We introduce quaternary modified four \(\mu \)circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four \(\mu \)circulant Hermitian selfdual codes. We also construct quaternary modified four \(\mu \)circulant Hermitian selfdual codes having large minimum weights. Two quaternary Hermitian selfdual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.
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1 Introduction
Selfdual codes are one of the most interesting classes of (linear) codes. This interest is justified by many combinatorial objects and algebraic objects related to selfdual codes (see e.g., [6, 20] and [26]).
Let \(\mathbb {F}_{q^2}\) denote the finite field of order \({q^2}\), where q is a prime or a prime power. A code C over \(\mathbb {F}_{q^2}\) of length n is said to be Hermitian selfdual if \(C=C^{\perp _H}\), where the Hermitian dual code \(C^{\perp _H}\) of C is defined as \(C^{\perp _H}=\{x \in \mathbb {F}_{q^2}^{n} \mid \langle x,y\rangle _H=0 \text { for all } y\in C\}\) under the Hermitian inner product \(\langle x,y\rangle _H\). By the Gleason–Pierce theorem, there are nontrivial divisible Hermitian selfdual codes over \(\mathbb {F}_{q^2}\) for \(q=2\) only. This is one of the reasons why much work has been done concerning Hermitian selfdual codes over \(\mathbb {F}_4\) (see e.g., [1, 4, 5, 9,10,11, 16,17,18,19, 21,22,25] and [27]). In this paper, we study Hermitian selfdual codes over \(\mathbb {F}_4\).
It is a fundamental and challenging problem in selfdual codes to classify selfdual codes and determine the largest minimum weight among all selfdual codes for a fixed length. A code over \(\mathbb {F}_4\) is called quaternary. All quaternary Hermitian selfdual codes were classified in [5, 16, 17] and [25] for lengths \(n \le 20\). Also, the largest minimum weight d(n) among all Hermitian selfdual codes is determined for lengths \(n \le 30\) (see [9, Table 5] for the current information on d(n)).
For small fields \(\mathbb {F}\), many four circulant (negacirculant) selfdual codes over \(\mathbb {F}\) having large minimum weights are known (see e.g., [7, 13,14,15] and the references given therein). In this paper, by modifying four circulant selfdual codes, we give a method for constructing quaternary Hermitian selfdual codes based on \(\mu \)circulant matrices, which are called modified four \(\mu \)circulant codes. Some basic properties of modified four \(\mu \)circulant quaternary Hermitian selfdual codes are given. We also give numerical results of quaternary modified four \(\mu \)circulant Hermitian selfdual codes together with an application to quantum codes.
This paper is organized as follows. In Sect. 2, we give some definitions, notations and basic results used in this paper. In Sect. 3, we define quaternary modified four \(\mu \)circulant codes as a certain modification of four circulant codes. We also give basic properties of quaternary modified four \(\mu \)circulant Hermitian selfdual codes. In particular, we give a condition for quaternary modified four \(\mu \)circulant codes to be Hermitian selfdual. In addition, we observe equivalences of quaternary modified four \(\mu \)circulant Hermitian selfdual codes. In Sect. 4, we present numerical results of quaternary modified four \(\mu \)circulant Hermitian selfdual codes. By computer search based on basic properties presented in Sect. 3, we give a classification of quaternary modified four \(\mu \)circulant Hermitian selfdual codes having the currently known largest minimum weights for lengths 24, 28, 32 and 36 (Proposition 7). For larger lengths, we also construct quaternary modified four \(\mu \)circulant Hermitian selfdual codes having large minimum weights. We emphasize that quaternary Hermitian selfdual [56, 28, 16] codes are constructed for the first time (Proposition 10). These codes \(C_{56,1}\) and \(C_{56,\omega }\) improve the previously known lower bounds on the largest minimum weight among all quaternary (linear) [56, 28] codes (Corollary 11). In Sect. 5, we give an application of \(C_{56,1}\) and \(C_{56,\omega }\) to quantum codes. More precisely, \(C_{56,1}\) and \(C_{56,\omega }\) imply the existence of a quantum [[56, 0, 16]] code.
2 Preliminaries
In this section, we give some definitions, notations and basic results used in this paper.
2.1 Quaternary codes
We denote the finite field of order 4 by \(\mathbb {F}_4=\{ 0,1,\omega , \overline{\omega }\}\), where \(\overline{\omega }= \omega ^2 = \omega +1\). A quaternary linear [n, k] code C is a kdimensional vector subspace of \(\mathbb {F}_4^n\). All codes in this paper are quaternary and linear unless otherwise noted, so we omit linear and we often omit quaternary. The parameter n is called the length of C. A generator matrix of C is a \(k \times n\) matrix such that the rows of the matrix generate C. The weight \({{\,\textrm{wt}\,}}(x)\) of a vector \(x \in \mathbb {F}_4^n\) is the number of nonzero components of x. The weight enumerator of C is given by \(\sum _{c \in C} y^{{{\,\textrm{wt}\,}}(c)}\). A vector of C is called a codeword of C. The minimum nonzero weight of all codewords in C is called the minimum weight of C. A quaternary [n, k, d] code is a quaternary [n, k] code with minimum weight d.
2.2 Quaternary Hermitian selfdual codes
The Hermitian dual code \(C^{\perp _H}\) of a quaternary code C of length n is defined as
under the following Hermitian inner product
for \(x=(x_1,x_2,\ldots ,x_n)\), \(y=(y_1,y_2,\ldots ,y_n)\in \mathbb {F}_{4}^n\). A quaternary code C is said to be Hermitian selfdual if \(C=C^{\perp _H}\). All codewords of a quaternary Hermitian selfdual code have even weights [25, Theorem 1].
All matrices in this paper are matrices over \(\mathbb {F}_4\), so we write simply matrices. Throughout this paper, let \(I_n\) denote the identity matrix of order n, and let \(A^T\) denote the transpose of a matrix A. Moreover, let \(\overline{A}\) denote the matrix \((a_{ij}^2)\) for a matrix \(A=(a_{ij})\). The following lemma is a criterion for Hermitian selfduality.
Lemma 1
[25, Theorems 1 and 4] Let C be a quaternary [2n, n] code with generator matrix \( \left( \begin{array}{cc} I_{n}&M \end{array} \right) \). If \(M \overline{M}^T=I_n\), then C is Hermitian selfdual.
It was shown in [25] that the minimum weight d of a quaternary Hermitian selfdual code of length n is bounded by:
A quaternary Hermitian selfdual code of length n and minimum weight \(2 \lfloor n/6 \rfloor +2\) is called extremal.
Two quaternary Hermitian selfdual codes C and \(C'\) are equivalent if there is a monomial matrix P over \(\mathbb {F}_4\) with \(C' = C \cdot P\), where \(C \cdot P = \{ x P\mid x \in C\}\) (see [25]). Throughout this paper, two equivalent quaternary Hermitian selfdual codes C and \(C'\) are denoted by \(C \cong C'\). All quaternary Hermitian selfdual codes were classified in [5, 16, 17] and [25] for lengths up to 20. All extremal quaternary Hermitian selfdual codes of length 22 were also classified in [17].
3 Definition and basic properties of modified four \(\mu \)circulant codes
In this section, we define quaternary modified four \(\mu \)circulant codes and we give their basic properties.
An \(n \times n\) matrix of the following form
is called \(\mu \)circulant, where \(\mu \in \{1,\omega ,\overline{\omega }\}\). In particular, if \(\mu =1\), then this is wellknown as a circulant matrix. It is trivial that a \(\mu \)circulant matrix with first row \((r_0,r_1,\ldots , r_{n1})\) is written as \(\sum _{i=0}^{n1} r_i E_n(\mu )^i\), where
Lemma 2
Suppose that \(\mu \in \{1,\omega ,\overline{\omega }\}\).

(i)
If A and B are \(n \times n\) \(\mu \)circulant matrices, then \(AB=BA\).

(ii)
If A is an \(n \times n\) \(\mu \)circulant matrix with first row \((r_0,r_1,\ldots , r_{n1})\), then \(\overline{A}^T\) is a \(\mu \)circulant matrix with first row \((r_0^2, (\mu r_{n1})^2,\ldots , (\mu r_1)^2)\).
Proof
The assertion (i) follows from the fact that a \(\mu \)circulant matrix with first row \((r_0, r_1,\ldots ,r_{n1})\) is written as \(\sum _{i=0}^{n1} r_i E_n(\mu )^i\). The assertion (ii) follows from the fact that \(\overline{A}^T\) is written as \(r_0^2I_n + \sum _{i=1}^{n1} (\mu r_{ni})^2 E_n(\mu )^i\). \(\square \)
By modifying four circulant selfdual codes (see e.g., [15] for the definition), we introduce the following method for constructing quaternary Hermitian selfdual codes. Suppose that \(\mu \in \{1,\omega ,\overline{\omega }\}\). Let A and B be \(n \times n\) \(\mu \)circulant matrices. We say that a quaternary [4n, 2n] code with generator matrix of the following form
is modified four \(\mu \)circulant. A modified four 1circulant code is also called modified four circulant. We denote the code with generator matrix (2) by \(C_{\mu }(A,B)\).
Remark 3
As a different modification of four circulant codes, codes with generator matrices of the following form
are given in [27], where A, B and C are circulant matrices and J is the exchange matrix.
Now we give some basic properties of modified four \(\mu \)circulant Hermitian selfdual codes. Although the following lemmas are somewhat trivial, we give proofs for the sake of completeness.
Lemma 4
Suppose that \(\mu \in \{1,\omega ,\overline{\omega }\}\). A quaternary modified four \(\mu \)circulant code \(C_{\mu }(A,B)\) is Hermitian selfdual if \(A \overline{A}^T +B \overline{B}^T =I_n\).
Proof
By Lemma 2 (i), \(AB+BA=O_n\), where \(O_n\) denotes the \(n \times n\) zero matrix. By Lemma 2 (ii), \(\overline{A}^T\) and \(\overline{B}^T\) are \(\mu \)circulant. Again by Lemma 2 (i), \(\overline{A}^T \overline{B}^T=\overline{B}^T\overline{A}^T\) and \(A\overline{A}^T =\overline{A}^TA\). Thus, we have
Let M(A, B) denote the \(2n \times 2n\) matrix \( \left( \begin{array}{cc} A &{} B \\ \overline{B}^T &{} \overline{A}^T \end{array} \right) \). Then we have
The result follows from Lemma 1. \(\square \)
Lemma 5
Suppose that \(C_{\mu }(A,B)\) is a quaternary modified four \(\mu \)circulant Hermitian selfdual code, where \(\mu \in \{1,\omega ,\overline{\omega }\}\). Then the following statements hold.

(i)
\(C_{\mu }(A,B) \cong C_{\mu }(\omega A,\omega B) \cong C_{\mu }(\overline{\omega }A,\overline{\omega }B)\).

(ii)
\(C_{\mu }(A,B) \cong C_{\mu }(B,A)\).

(iii)
\(C_{\mu }(A,B) \cong C_{\mu }(\overline{A}^T, \overline{B}^T)\).

(iv)
\(C_{\mu }(A,B) \cong C_{\mu }(A,\overline{B}^T)\).
Proof
The assertions (i), (ii) and (iii) are trivial. The Hermitian dual code \(C_{\mu }(A,B)^{\perp _H}\) of \(C_{\mu }(A,B)\) has the following generator matrix
Since \(C_{\mu }(A,B)=C_{\mu }(A,B)^{\perp _H}\), the above matrix is also a generator matrix of \(C_{\mu }(A,B)\). It follows from (iii) that \(C_{\mu }(A,B) \cong C_{\mu }(A,\overline{B}^T)\). \(\square \)
Lemma 6
Let C be a quaternary modified four \(\mu \)circulant Hermitian selfdual code, where \(\mu \in \{1,\omega ,\overline{\omega }\}\). Then there is a quaternary modified four \(\mu \)circulant Hermitian selfdual code \(C_{\mu }(A,B)\) such that \(C \cong C_{\mu }(A,B)\) and the first nonzero coordinate of the first row of A is 1.
Proof
Suppose that \(C=C_{\mu }(A',B')\) and the first nonzero coordinate of the first row of \(A'\) is \(\omega \) (resp. \(\overline{\omega }\)). Then \(C_{\mu }(\overline{\omega }A', \overline{\omega }B')\) (resp. \(C_{\mu }(\omega A', \omega B')\)) is a modified four \(\mu \)circulant code such that nonzero coordinate of the first row of \(\overline{\omega }A'\) (resp. \(\omega A'\)) is 1. By Lemma 5 (i), we have that \(C \cong C_{\mu }(\overline{\omega }A', \overline{\omega }B')\) (resp. \(C \cong C_{\mu }(\omega A', \omega B')\)). The result follows. \(\square \)
The above lemma substantially reduces the number of codes which need be checked when a classification of modified four \(\mu \)circulant Hermitian selfdual codes is completed and the largest minimum weight among all modified four \(\mu \)circulant Hermitian selfdual codes is determined in the next section.
4 Numerical results of modified four \(\mu \)circulant Hermitian selfdual codes
In this section, we present numerical results of quaternary modified four \(\mu \)circulant Hermitian selfdual codes. We emphasize that Hermitian selfdual [56, 28, 16] codes are constructed. These codes are the first examples of not only Hermitian selfdual [56, 28, 16] codes but also (linear) [56, 28, 16] codes. All computer calculations in this section were done using programs in Magma [2] unless otherwise specified.
4.1 Classification of modified four \(\mu \)circulant Hermitian selfdual codes
As described in Sect. 2, all quaternary Hermitian selfdual codes of lengths up to 20 were classified in [5, 16, 17] and [25]. From now on, we consider Hermitian selfdual codes for only lengths \(n \ge 24\).
Let d(n) denote the largest minimum weight among all Hermitian selfdual codes of length n. Let \(d^K(n)\) denote the largest minimum weight among previously known Hermitian selfdual codes of length n. For \(n \in \{24,28,\ldots ,80\}\), the values \(d^K(n)\) are listed in Table 1, noting that \(d(24)=8\) and \(d(28)=10\) (see [9, Table 5]).
Here we give a classification of modified four \(\mu \)circulant Hermitian selfdual codes having minimum weight \(d^K(n)\) for length \(n\in \{24,28,32,36\}\). We describe how to complete our classification briefly. Our exhaustive computer search based on Lemmas 4 and 6 found all distinct generator matrices (2) of modified four \(\mu \)circulant Hermitian selfdual \([n,n/2,d^K(n)]\) codes \(C_{\mu }(A,B)\), which must be checked further for equivalences. To test equivalence of two modified four \(\mu \)circulant Hermitian selfdual \([n,n/2,d^K(n)]\) codes, we used Magma function IsIsomorphic. Moreover, in the process of finding these codes, we verified that there is no modified four \(\mu \)circulant Hermitian selfdual code of length n and minimum weight \(d >d^K(n)\) for lengths \(n=32\) and 36. Then we have the following proposition.
Proposition 7

(i)
Up to equivalence, there are 7 quaternary modified four circulant Hermitian selfdual [24, 12, 8] codes. Up to equivalence, there are 9 quaternary modified four \(\mu \)circulant Hermitian selfdual [24, 12, 8] codes for \(\mu \in \{\omega ,\overline{\omega }\}\).

(ii)
Up to equivalence, there are 3 quaternary modified four \(\mu \)circulant extremal Hermitian selfdual [28, 14, 10] codes for \(\mu \in \{1,\omega ,\overline{\omega }\}\).

(iii)
Up to equivalence, there are 59 quaternary modified four \(\mu \)circulant Hermitian selfdual [32, 16, 10] codes for \(\mu \in \{1,\omega ,\overline{\omega }\}\). If \(d \ge 12\), then there is no quaternary modified four \(\mu \)circulant Hermitian selfdual [32, 16, d] code for \(\mu \in \{1,\omega ,\overline{\omega }\}\).

(iv)
Up to equivalence, there is a unique quaternary modified four \(\mu \)circulant Hermitian selfdual [36, 18, 12] code for \(\mu \in \{1,\omega ,\overline{\omega }\}\). If \(d\ge 14\), then there is no quaternary modified four \(\mu \)circulant Hermitian selfdual [36, 18, d] code for \(\mu \in \{1,\omega ,\overline{\omega }\}\).
For \(n \in \{24,28,32,36\}\), by \(C_{n,\mu ,i}\) \((i \in \{1,2,\ldots ,N_\mu (n)\})\), we denote the modified four \(\mu \)circulant Hermitian selfdual \([n,n/2,d^K(n)]\) codes described in the above proposition, where
For these codes \(C_{n,\mu ,i}=C_{\mu }(A,B)\) \((\mu =1,\omega ,\overline{\omega })\), the first rows \(r_A\) (resp. \(r_B\)) of A (resp. B) are listed in Tables 2, 3, 4, 9, 10 and 11.
Remark 8
By Magma function IsIsomorphic, we have the following
and there is no other pair of equivalent codes among the codes described in Proposition 7.
For \(n=24,32\) and 36, the possible weight enumerators of quaternary Hermitian selfdual \([n,n/2,d^K(n)]\) codes can be written using \(A_{d^K(n)}\) (see [1] and [22, Sect. III]). Note that the possible weight enumerator of an extremal Hermitian selfdual code of a fixed length is uniquely determined. For the above codes \(C_{n,\mu ,i}\) (\(n=24,32\) and 36), the numbers \(A_{d^K(n)}\) of codewords of minimum weight \(d^K(n)\) are also listed in Tables 2, 4, 9, 10 and 11. This was calculated by the Magma function NumberOfWords.
4.2 Largest minimum weights of modified four \(\mu \)circulant Hermitian selfdual codes
We give some observations on the largest minimum weight d(n) among all Hermitian selfdual codes of length n and the largest minimum weight \(d_{\mu }(n)\) \((\mu =1,\omega ,\overline{\omega })\) among all modified four \(\mu \)circulant Hermitian selfdual codes of length n. For lengths \(n=40\) and 44, by a method similar to the above, our exhaustive computer search based on Lemmas 4 and 6 verified that there is no modified four \(\mu \)circulant Hermitian selfdual [n, n/2, d] code with \(d > d^K(n)\) for \(\mu \in \{1,\omega ,\overline{\omega }\}\) (see Table 1 for the minimum weights \(d^K(n)\)). In addition, we found a modified four \(\mu \)circulant Hermitian selfdual \([n,n/2,d^K(n)]\) code \(C_{n,\mu }\) for \(\mu \in \{1,\omega ,\overline{\omega }\}\). This implies the following proposition.
Proposition 9
For \(\mu \in \{1,\omega ,\overline{\omega }\}\), \(d_{\mu }(40)=10\text { and } d_{\mu }(44)=12\).
For the above codes \(C_{40,\mu }=C_{\mu }(A,B)\) and \(C_{44,\mu }=C_{\mu }(A,B)\), the first rows \(r_A\) (resp. \(r_B\)) of A (resp. B) are listed in Table 5. The numbers \(A_{d^K(n)}\) of codewords of minimum weight \(d^K(n)\) are also listed in the table. This was calculated by the Magma function NumberOfWords. The numbers show that these codes are inequivalent.
For lengths \(48,52,\ldots ,76\) and 80, by a nonexhaustive search based on Lemmas 4 and 6, we continued finding modified four \(\mu \)circulant Hermitian selfdual codes having large minimum weights. Then we found a modified four \(\mu \)circulant Hermitian selfdual code \(C_{n,\mu }\) of length n and minimum weight d for
For the above codes \(C_{n,\mu }=C_{\mu }(A,B)\), the first rows \(r_A\) (resp. \(r_B\)) of A (resp. B) are listed in Table 12. We have the following proposition.
Proposition 10
There are quaternary Hermitian selfdual [56, 28, 16] codes.
We emphasize that \(C_{56,1}\) and \(C_{56,\omega }\) are the first examples of not only Hermitian selfdual [56, 28, 16] codes but also (linear) [56, 28, 16] codes [8]. We give the weight enumerators of these codes in the next subsection.
In Table 6, we summarize the current information on \(d_1(n)\), \(d_\omega (n)\) and \(d_{\overline{\omega }}(n)\). The upper bounds on \(d_1(n)\), \(d_\omega (n)\) and \(d_{\overline{\omega }}(n)\) follow from (1). The lower bounds on \(d_1(n)\), \(d_\omega (n)\) and \(d_{\overline{\omega }}(n)\) follow from Table 12.
4.3 \(C_{56,1}\) and \(C_{56,\omega }\)
It is a main problem in coding theory to determine the largest minimum weight \(d_q(n,k)\) among all [n, k] codes over a finite field of order q for a given (q, n, k). The current information on \(d_4(n,k)\) can be found in [8]. For example, it was previously known that \(15 \le d_4(56,28) \le 21\). As a consequence of Proposition 10, we have the following corollary.
Corollary 11
\(16 \le d_4(56,28) \le 21\).
Now we determine the weight enumerators of \(C_{56,1}\) and \(C_{56,\omega }\). It is well known that the possible weight enumerators of quaternary Hermitian selfdual codes can be determined by the Gleason type theorem [24, p. 804] (see also [25, Theorem 13]). The weight enumerator W of a quaternary Hermitian selfdual code of length n is written as:
using some integers \(a_j\). The possible weight enumerator \(W_{56,16}=\sum _{i=0}^{56}A_i y^i\) of a quaternary Hermitian selfdual [56, 28, 16] code is determined by (3), where \(A_i\) are listed in Table 7 together with \(\alpha =A_{16}\) and \(\beta =A_{18}\). Only this calculation was done by Mathematica [28]. By the Magma function NumberOfWords, we calculated that
for \(C_{56,1}\) and \(C_{56,\omega }\), respectively. This determines the weight enumerators of \(C_{56,1}\) and \(C_{56,\omega }\).
4.4 Largest minimum weights d(n)
In Table 8, we summarize the current information on the largest minimum weights d(n) for \(n \in \{24,28,\ldots ,80\}\). The upper bounds on d(n) follow from (1). The references about the lower bounds on d(n) are also listed in the table.
In [9, Table 5], the largest minimum weights d(n) were considered for \(n \le 80\). Here we investigate the largest minimum weights d(n) for \(n \in \{84, 88, 92, 96, 100\}\). A Hermitian selfdual code of length n and minimum weight 22 is given in [8] for \(n=92\) and 100. We denote the two codes by \(G_{92}\) and \(G_{100}\), respectively. As information, we briefly give the construction of \(G_{92}\) and \(G_{100}\). Let \(G_{91,1}\) and \(G_{91,2}\) denote the cyclic codes of length 91 with generator polynomials \(g_1\) and \(g_2\), respectively, where
The code \(G_{92}\) is constructed from \(G_{91,1}\), \(G_{91,2}\) and the [1, 1] code by Construction X. The code \(G_{100}\) is equivalent to the double circulant code with generator matrix \( \left( \begin{array}{cc} I_{50}&R \end{array} \right) \), where R is the circulant matrix with the first row
For \((n,d)=(84,20)\), (88, 20) and (96, 22), by a nonexhaustive search based on Lemmas 4 and 6, we found a modified four \(\omega \)circulant Hermitian selfdual code \(C_{n,\omega }=C_{\omega }(A,B)\) of length n and minimum weight d. For the above codes, the first rows \(r_A\) (resp. \(r_B\)) of A (resp. B) are listed in Table 12. In Table 6, we give lower and upper bounds on the largest minimum weights d(n) for \(n \in \{84, 88, 92, 96, 100\}\). The upper bounds on d(n) follow from (1). The references about the lower bounds on d(n) are also listed in the table. For
a Hermitian selfdual code of length n and minimum weight d is constructed for the first time. In Table 8, the minimum weights of these codes are given in bold.
5 Application to quantum codes
In this section, we consider an application of the quaternary Hermitian selfdual [56, 28, 16] codes \(C_{56,1}\) and \(C_{56,\omega }\) found in the previous section to quantum codes.
A quaternary additive \((n,2^k)\) code \(\mathcal {C}\) is an additive subgroup of \(\mathbb {F}_4^n\) with \(\mathcal {C}=2^k\). The dual code \({\mathcal {C}}^*\) of a quaternary additive \((n,2^k)\) code \(\mathcal {C}\) is defined as
under the following trace inner product
for \(x=(x_1,x_2,\ldots ,x_n)\), \(y=(y_1,y_2,\ldots ,y_n) \in \mathbb {F}_4^n\). A quaternary additive code \(\mathcal {C}\) is called selforthogonal and selfdual if \(\mathcal {C}\subset {\mathcal {C}}^*\) and \(\mathcal {C}= {\mathcal {C}}^*\), respectively. Note that a quaternary Hermitian selfdual [n, n/2, d] code is a quaternary additive selfdual \((n,2^n)\) code with minimum weight d (see e.g., [12]).
A useful method for constructing quantum codes from quaternary additive selforthogonal codes was given by Calderbank, Rains, Shor and Sloane [3] (see [3] for undefined terms concerning quantum codes). A quaternary additive selforthogonal \((n,2^{nk})\) code \(\mathcal {C}\) such that there is no vector of weight less than d in \({\mathcal {C}}^* \setminus \mathcal {C}\), gives a quantum [[n, k, d]] code, where \(k \ne 0\). A quaternary additive selfdual \((n,2^n)\) code with minimum weight d gives a quantum [[n, 0, d]] code. Let \(d_{\max }(n,k)\) denote the largest minimum weight d among quantum [[n, k, d]] codes. Similar to the classical coding theory, it is a fundamental problem to determine \(d_{\max }(n,k)\). A table on \(d_{\max }(n,k)\) is given in [3, Table III] for \(n \le 30\). An extended table is obtained electronically from [8]. For example, it was previously known that \(15 \le d_{\max }(56,0) \le 20\) [8].
In Sect. 4, quaternary Hermitian selfdual [56, 28, 16] codes \(C_{56,1}\) and \(C_{56,\omega }\) were constructed for the first time. By the above method, a quantum [[56, 0, 16]] code is obtained. Hence, we have the following proposition.
Proposition 12

(i)
There is a quantum [[56, 0, 16]] code.

(ii)
\(16 \le d_{\max }(56,0) \le 20\).
Data availibility
Available from the corresponding author on reasonable request.
Code availability
Not applicable.
References
Araya M., Harada M.: Some restrictions on the weight enumerators of nearextremal ternary selfdual codes and quaternary Hermitian selfdual codes. Des. Codes Cryptogr. 91, 1813–1843 (2023).
Bosma W., Cannon J., Playoust C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997).
Calderbank A.R., Rains E.R., Shor P.W., Sloane N.J.A.: Quantum error correction via codes over \(\rm GF (4)\). IEEE Trans. Inform. Theory 44, 1369–1387 (1998).
Conway J.H., Pless V.: Monomials of orders \(7\) and \(11\) cannot be in the group of a \((24,12,10)\) selfdual quaternary code. IEEE Trans. Inform. Theory 29, 137–140 (1983).
Conway J.H., Pless V., Sloane N.J.A.: Selfdual codes over \(GF(3)\) and \(GF(4)\) of length not exceeding \(16\). IEEE Trans. Inform. Theory 25, 312–322 (1979).
Conway J.H., Sloane N.J.A.: Sphere Packing, Lattices and Groups, 3rd edn Springer, New York (1999).
Georgiou S.D., Lappas E.: Selfdual codes from circulant matrices. Des. Codes Cryptogr. 64, 129–141 (2012).
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de/. Accessed 24 Nov 2022.
Grassl M., Gulliver T.A.: On circulant selfdual codes over small fields. Des. Codes Cryptogr. 52, 57–81 (2009).
Gulliver T.A.: Optimal double circulant selfdual codes over \(\mathbb{F} _4\). IEEE Trans. Inform. Theory 46, 271–274 (2000).
Gulliver T.A., Harada M., Miyabayashi H.: Optimal double circulant selfdual codes over \(\mathbb{F} _4\) II. Australas. J. Comb. 39, 163–174 (2007).
Gulliver T.A., Kim J.L.: Circulant based extremal additive selfdual codes over \(\rm GF (4)\). IEEE Trans. Inform. Theory 50, 359–366 (2004).
Harada M.: New doubly even selfdual codes having minimum weight \(20\). Adv. Math. Commun. 14, 89–96 (2020).
Harada M.: Selfdual codes over \(\mathbb{F} _5\) and \(s\)extremal unimodular lattices. Discret. Math. 346, 113126 (2023).
Harada M., Holzmann W., Kharaghani H., Khorvash M.: Extremal ternary selfdual codes constructed from negacirculant matrices. Gr. Comb. 23, 401–417 (2007).
Harada M., Lam C., Munemasa A., Tonchev V.D.: Classification of generalized Hadamard matrices \(H(6,3)\) and quaternary Hermitian selfdual codes of length \(18\). Electron. J. Comb. 17, Research Paper 171 (2010).
Harada M., Munemasa A.: Classification of quaternary Hermitian selfdual codes of length \(20\). IEEE Trans. Inform. Theory 57, 3758–3762 (2011).
Huffman W.C.: On extremal selfdual quaternary codes of lengths \(18\) to \(28\) I. IEEE Trans. Inform. Theory 36, 651–660 (1990).
Huffman W.C.: On extremal selfdual quaternary codes of lengths \(18\) to \(28\) II. IEEE Trans. Inform. Theory 37, 1206–1216 (1991).
Huffman W.C.: On the classification and enumeration of selfdual codes. Finite Fields Appl. 11, 451–490 (2005).
Kim H.J., Lee Y.: Classification of extremal selfdual quaternary codes of lengths \(30\) and \(32\). IEEE Trans. Inform. Theory 59, 2352–2358 (2013).
Kim J.L.: New selfdual codes over \(\rm GF (4)\) with the highest known minimum weights. IEEE Trans. Inform. Theory 47, 1575–1580 (2001).
Lam C.W.H., Pless V.: There is no \((24,12,10)\) selfdual quaternary code. IEEE Trans. Inform. Theory 36, 1153–1156 (1990).
MacWilliams F.J., Mallows C.L., Sloane N.J.A.: Generalizations of Gleason’s theorem on weight enumerators of selfdual codes. IEEE Trans. Inform. Theory 18, 794–805 (1972).
MacWilliams F.J., Odlyzko A.M., Sloane N.J.A., Ward H.N.: Selfdual codes over \(\rm GF (4)\). J. Comb. Theory Ser. A 25, 288–318 (1978).
Rains E., Sloane N.J.A.: Selfdual codes. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998).
Roberts A.M.: Quaternary Hermitian selfdual codes of lengths \(26\), \(32\), \(36\), \(38\) and \(40\) from modifications of wellknown circulant constructions. Appl. Algebra Eng. Commun. Comput. (to appear). https://doi.org/10.1007/s0020002200589w.
Wolfram Research, Inc., Mathematica, Version 12.3.1. https://www.wolfram.com/mathematica.
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This work was supported by JSPS KAKENHI Grant Number 19H01802.
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Harada, M. A method for constructing quaternary Hermitian selfdual codes and an application to quantum codes. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s1062302401421x
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DOI: https://doi.org/10.1007/s1062302401421x