Quaternary Hermitian Linear Complementary Dual Codes

Abstract

The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension 2. In this paper, we give some conditions on the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum weights. As a consequence, we completely determine the largest minimum weights for dimension 3, by using a classification of some quaternary codes. In addition, for a positive integer s, an entanglement-assisted quantum error-correcting [[21s+5, 3, 16s+3; 21s+2]] code with maximal entanglement is constructed for the first time from a quaternary Hermitian linear complementary dual [26,3, 19] code. Recently, much work has been done concerning LCD codes for both theoretical and practical reasons (see e.g. [1], [5], [6], [8], [11], [12] and the references given therein). For example, if there is a quaternary Hermitian LCD [n, k, d] code, then there is a maximal entanglement [[n, k, d; n − k]] entanglementassisted quantum error-correcting code (EAQECC for short) (see e.g. [11] and [12]). From this point of view, quaternary Hermitian LCD codes play an important role in the study of maximal entanglement EAQECC's.It is a fundamental problem to determine the largest minimum weight d 4 (n, k) among all quaternary Hermitian LCD [n, k] codes for a given pair (n, k). It was shown that d 4 (n, 2) = ⌊ 4n 5 ⌋ if n ≡ 1, 2, 3 (mod 5) and d 4 (n, 2) = ⌊ 4n 5 ⌋ − 1 if n ≡ 0, 4 (mod 5) for n ≥ 3 in [10] and [12]. In this paper, we give some conditions on the nonexistence of quaternary Hermitian LCD codes with large minimum weights. We give a classification of (unrestricted) quaternary [4r, 3, 3r] codes for r = 9, 10, 12, 13, 14, 16 and quaternary [43, 3, 32] codes. Using the above classification and the classification in [3], we completely determine the largest minimum weight among all quaternary Hermitian LCD codes of dimension 3. In addition, for a positive integer s, a maximal entanglement [[21s + 5, 3, 16s + 3; 21s + 2]] EAQECC is constructed for the first time from a quaternary Hermitian LCD [26,3, 19] code. This paper is organized as follows. In Section 2, we prepare some definitions, notations and basic results used in this paper. In Section 3, we give characterizations of quaternary Hermitian LCD codes. It is shown that there is no quaternary Hermitian LCD [ 4 k −1 3 s, k, 4 k−1 s] code for k ≥ 3 and s ≥ 1 (Theorem 3.3). In addition, if 4(4 k−1 n − 4 k −1 3 α) < k, where k ≥ 3 and 4α − 3n ≥ 1, then there is no quaternary Hermitian LCD [n, k, α] code C with d(C ⊥ H ) ≥ 2, where d(C) denotes the minimum (Hamming) weight of a quaternary code C and C ⊥ H denotes the Hermitian dual code of C. If 4(4 k−1 n − 4 k −1 3 α) ≥ k ≥ 3, where 4α − 3n ≥ 1 and there is no quaternary Hermitian LCD [4(4 k−1 n − 4 k −1 3 α), k, 3(4 k−1 n − 4 k −1 3 α)] code C 0 with d(C ⊥ H 0 ) ≥ 2, then there is no quaternary Hermitian LCD [n, k, α] code C with d(C ⊥ H ) ≥ 2 (Theorem 3.4). In Section 4, from the classification of quaternary codes of dimension 3 by Bouyukliev, Grassl and Varbanov [3], we determine...

On the classification of quaternary optimal Hermitian LCD codes

“…By following the same line as in the proof of Theorem 9 in [2], we have the following theorem. ) ∈ Z…”

“…The largest minimum weights d 4 (n, 2) were determined in [2], where d 4 (n, 2) are listed in Table 1. Recently, Ishizuka [8] has completed a classification of quaternary optimal Hermitian LCD codes of dimension 2.…”

“…It was shown that d 4 (n, 2) = ⌊ 4n 5 ⌋ if n ≡ 1, 2, 3 (mod 5) and d 4 (n, 2) = ⌊ 4n 5 ⌋ − 1 otherwise for n ≥ 3 [10] and [11]. Recently, it has been shown that d 4 (n, 3) = ⌊ 16n 21 ⌋ if n ≡ 5, 9, 13, 17, 18 (mod 21) and d 4 (n, 3) = ⌊ 16n 21 ⌋ − 1 otherwise for n ≥ 6 [2] and [11]. More recently, Ishizuka [8] has completed a classification of quaternary optimal Hermitian LCD codes of dimension 2.…”

“…Araya, Harada and Saito [2] gave some conditions on the nonexistence of certain quaternary Hermitian LCD codes having large minimum weights ([2, Theorem 9]). In this note, by following the same line as in the proof of [2,Theorem 9], we propose a method for a classification of quaternary Hermitian LCD codes having large minimum weights.…”

“…Araya, Harada and Saito [2] gave some conditions on the nonexistence of certain quaternary Hermitian LCD codes having large minimum weights ([2, Theorem 9]). In this note, by following the same line as in the proof of [2,Theorem 9], we propose a method for a classification of quaternary Hermitian LCD codes having large minimum weights. By using this method, we complete a classification of quaternary optimal Hermitian LCD [n, 3, d 4 (n, 3)] codes for arbitrary n. We also give an alternative classification of quaternary optimal Hermitian LCD [n, 2, d 4 (n, 2)] codes and a classification of quaternary near-optimal Hermitian LCD [n, 2, d 4 (n, 2) − 1] codes for arbitrary n.…”