# Classification of optimal quaternary Hermitian LCD codes of dimension 2

Hermitian linear complementary dual codes are linear codes which intersect with the Hermitian dual codes trivially. The largest minimum weight among quaternary Hermitian linear complementary dual codes of dimension 2 is known for each length. We give the complete classification of optimal quaternary Hermitian linear complementary dual codes of dimension 2.

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## 1 Introduction

Let be the finite field of order four, where satisfies . The conjugate of is defined as . A quaternary code is a linear subspace of with dimension and minimum weight . Throughout this paper, we consider only linear quaternary codes and omit the term “linear quaternary”. Given a code

, a vector

is said to be a codeword of . The weight of a codeword is denoted by .

Let be vectors of . The Hermitian inner product is defined as . Given a code , the Hermitian dual code of is . A generator matrix of the code is any matrix whose rows form a basis of . Moreover, a generator matrix of the Hermitian dual code is said to be a parity check matrix of . Given a matrix , we denote the transpose of by and the conjugate of by . Hermitian linear complementary dual codes, Hermitian LCD codes for short, are codes whose intersection with the Hermitian dual codes are trivial. The concept of LCD codes was invented by Massey [7] in 1992. LCD codes have been applied in data storage, communication systems and cryptography. For example, it is known that LCD codes can be used against side-channel attacks and fault injection attacks [3]. We note that a code is a Hermitian LCD code if and only if its generator matrix satisfies  [5].

Two codes are equivalent if one can be obtained from the other by a permutation of the coordinates and a multiplication of any coordinate by a nonzero scalar. We denote the equivalence of two codes by . Let be generator matrices of two codes respectively. It is known that if and only if can be obtained from by an elementary row operation, a permutation of the columns and multiplication of any column by a nonzero scalar.

It was shown in [6] that the upper bound of the minimum weight of the Hermitian LCD codes is given as follows:

 d≤⎧⎪⎨⎪⎩⌊4n5⌋if n≡1,2,3(mod5),⌊4n5⌋−1otherwise. (1)

Also, it was proved that for all , there exists a Hermitian LCD code which meets this upper bound. We say that a Hermitian LCD code is optimal if it meets this upper bound. It was shown in [4] that any code over is equivalent to some Hermitian LCD code for . Furthermore, it was proved in [6] that a Hermitian LCD code constructs a maximal-entanglement entanglement-assisted quantum error correcting code. Motivated by the results, we are concerned with the complete classification of optimal Hermitian LCD codes of dimension .

This paper is organized as follows. In Section 2, we present a method to construct optimal Hermitian LCD codes of dimension

, including all inequivalent codes. Also, a method to classify optimal Hermitian LCD codes of dimension

is given. In Section 3, we classify optimal Hermitian LCD codes of dimension . Up to equivalence, the complete classification of optimal Hermitian LCD codes of dimension is given. It is shown that all inequivalent codes have distinct weight enumerators, which is used for the classification.

## 2 Classification Method

Let be the zero vector of length and be the all-ones vector of length . Let be a tuple of nonnegative integers. We introduce the following notation:

 G(a0,a1,a2,a3,a4,a5)\coloneqq(100a00a11a21a31a41a5010a01a10a21a3ω1a4ω21a5).

We denote by the code whose generator matrix is . By the same argument as in [1], we obtain the following lemma.

###### Lemma 1.

Given a code , define . Let denote the set of all inequivalent Hermitian LCD codes C such that the minimum weight of is . Then there exists a set of all inequivalent Hermitian LCD codes such that .

We assume by Lemma 1 and omit . Furthermore, throughout this paper, we use the following notations:

 G(a)\coloneqqG(a1,a2,a3,a4,a5), C(a)\coloneqqC(a1,a2,a3,a4,a5), (2)

respectively, to save space.

###### Proposition 1.

Let be an code. Then there exist nonnegative integers such that and .

###### Proof.

Let be a generator matrix of the code . By multiplying rows by some non-zero scalars, is changed to a generator matrix which consists only of the columns of . Permuting the columns, is obtained from . Hence it holds that . Since the minimum weight of is , we may assume that the first row of is a codeword with weight , which yields . ∎

Given a code , we may assume without loss of generality that satisfies

 1+a2+a3+a4+a5=d, (3)

by Proposition 1. This assumption on a generator matrix reduces computations later.

###### Lemma 2.

Let be an code . Then is a Hermitian LCD code if and only if satisfies the following conditions:

 a1=n−d−1, (4) a2≤n−d−1, (5) a3,a4,a5≤n−d, (6) {a3+a4+a5+a3a4+a4a5+a5a3≢0(mod2)ifdiseven,a3a4+a4a5+a5a3≢n−d(mod2)otherwise. (7)
###### Proof.

Suppose is a Hermitian LCD code. Let be a generator matrix of the code . Let be the first and second rows of respectively. The number of columns of equals to the length . Thus, it holds that

 2+a1+a2+a3+a4+a5=n. (8)

Since the minimum weight of is , the following holds: . By (3), we have . Substituting (8) in each equation, we obtain (4) through (6).

The code is a Hermitian LCD code if and only if

 detG¯¯¯¯GT=(r1,r1)h(r2,r2)h−(r2,r1)h(r1,r2)h≠0,

where

 (r1,r1)h =1+a2+a3+a4+a5=d, (r1,r2)h =a3+ωa5+ω2a4=a3+a4+ω(a4+a5), (r2,r1)h =a3+ωa4+ω2a5=a3+a5+ω(a4+a5), (r2,r2)h =1+a1+a3+a4+a5.

Here we regard are elements of . Therefore, (4) through (7) hold if is a Hermitian LCD code and vice versa. ∎

Given an code , we define the following:

 bi \coloneqq(n−d)−ai for 1≤i≤5. (9)
###### Lemma 3.

Let be the code with the parameters . Then is a Hermitian LCD code if and only if satisfies the following conditions:

 b1=1, (10) b2≥1, (11) b3,b4,b5≥0, {b3+b4+b5+b3b4+b4b5+b5b3≠0ifdiseven,b3b4+b4b5+b5b3≠0otherwise.
###### Proof.

The result follows from Lemma 2. ∎

Given an code , we define the following:

 Δ \coloneqq4n−5d. (12)
###### Lemma 4.

Let be a Hermitian LCD code. Then is optimal if and only if the value of with respect to is given as follows:

 Δ=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩5if n≡0(mod5),4if n≡1(mod5),3if n≡2(mod5),2if n≡3(mod5),6if n≡4(mod5).
###### Proof.

The result follows from (12). ∎

###### Lemma 5.

Let be a code . If is a Hermitian LCD code, then it holds that

 0≤b3,b4,b5≤Δ.
###### Proof.

Substituting in (3), we obtain

 b2=Δ+1−(b3+b4+b5). (13)

Combining with (11), we obtain . Since , it follows that . ∎

###### Lemma 6.
 C(a)≃C(a1,a2,a5,a3,a4). (14)
###### Proof.

Multiply the second row of by . Permuting the columns, the result follows. Note that we may assume that the nonzero entry of a column is , provided that the entry of the other column is . ∎

By Lemma 6, we may assume . Notice that if and only if by (9).

## 3 Optimal Hermitian LCD Codes of Dimension 2

By Lemmas 3 through 6, it suffices to calculate all satisfying

 0≤b3≤b4,b5≤Δ, (15) {b3+b4+b5+b3b4+b4b5+b5b3≠0ifdiseven,b3b4+b4b5+b5b3≠0otherwise, (16)

in order to obtain optimal Hermitian LCD codes of dimension , including all inequivalent codes. Notice that are obtained by (10), (13) respectively. Our computer search found all integers satisfying (15) and (16). This calculation was done by Magma [2]. Recall that are obtained from by (9). For optimal Hermitian LCD codes of dimension , the integers are listed in Table 1, where the rows are in lexicographical order with respect to , and is a nonnegative integer.

###### Lemma 7.

Suppose is a positive integer. Then

 C(a)≃C(a1,a3−1,a2+1,a5,a4). (17)
###### Proof.

Add the second row of to the first row. Permuting the columns, the result follows. Recall that is defined in (2). ∎

###### Lemma 8.

Suppose . Then

 C(a)≃C(a2,a1,a3,a5,a4). (18)
###### Proof.

Interchange the first row and the second row of . Permuting the columns, the result follows. ∎

By Lemmas 6 through 8, we have the equivalences among some codes listed in Table 1, which are displayed in Table 2. Note that denotes the two codes are equivalent by . Also, denotes that, given two codes , there exists a code such that .

Table 3 gives the weight enumerators of representatives in Table 2. The weight enumerator is given by , where be the first and second rows of respectively. Since the weight enumerators are distinct, the codes in Table 3 are inequivalent. Table 4 gives the classification of optimal Hermitian LCD codes of dimension , with the case where is so small that some codes in Table 1 do not exist.

Recall that we have assumed by Lemma 1. It follows from (1) that there exists an optimal Hermitian LCD code such that the minimum weight of equals to if and only if . Consequently, we obtain the following theorem.

###### Theorem 1.
1. [label=]

2. Up to equivalence, there exist two optimal Hermitian LCD codes for every integer with .

3. Up to equivalence, there exist two optimal Hermitian LCD codes for every integer with .

4. Up to equivalence, there exists unique optimal Hermitian LCD codes for every integer with .

5. Up to equivalence, there exists unique optimal Hermitian LCD codes for every integer with .

6. Up to equivalence, there exist six optimal Hermitian LCD codes for every integer with . One of them is the code such that the minimum weight of the Hermitian dual code is .

## Acknowledgements

The author would like to thank supervisor Professor Masaaki Harada for introducing the problem, useful discussions and his encouragement.

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