Hermitian linear complementary dual codes are linear codes whose intersection
with their Hermitian dual code is trivial. 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 $2$

Ishizuka, Keita

English

arXiv

Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitian dual codes are trivial. 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 $2$

Journal of Algebra Combinatorics Discrete Structures and Applications (JACODESMATH, Yildiz Technical University - YTU)

ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.790748J. Algebra Comb. Discrete Appl.7(3) • 229–236Received: 18 April 2020Accepted: 28 May 2020Journal of Algebra Combinatorics Discrete Structures and ApplicationsClassification of optimal quaternary Hermitian LCDcodes of dimension 2Research ArticleKeita IshizukaAbstract: Hermitian linear complementary dual codes are linear codes whose intersections with their Hermitiandual codes are trivial. The largest minimum weight among quaternary Hermitian linear complemen-tary dual codes of dimension 2 is known for each length. We give the complete classification of optimalquaternary Hermitian linear complementary dual codes of dimension 2.2010 MSC: 94B05Keywords: Linear complementary dual code, Hermitian linear complementary dual code, Optimal codes1. IntroductionLet F4 := {0, 1, ω, ω2} be the finite field of order four, where ω satisfies ω2+ω+1 = 0. The conjugateof x ∈ F4 is defined as x := x2. A quaternary [n, k, d] code is a linear subspace of Fn4 with dimension kand minimum weight d. Throughout this paper, we consider only linear quaternary codes and omit theterm “linear quaternary”. Given a code C, a vector c ∈ C is said to be a codeword of C. The weight of acodeword c is denoted by wt(c).Let u := (u1, u2, . . . , un), v := (v1, v2, . . . , vn) be vectors of Fn4 . The Hermitian inner product isdefined as (u, v)h := u1v1 + u2v2 + · · · + unvn. Given a code C, the Hermitian dual code of C isC⊥h := {x ∈ Fn4 | (x, y)h = 0 for all y ∈ C}. A generator matrix of the code C is any matrix whoserows form a basis of C. Moreover, a generator matrix of the Hermitian dual code C⊥h is said to be aparity check matrix of C. Given a matrix G, we denote the transpose of G by GT and the conjugateof G by G. Hermitian linear complementary dual codes, Hermitian LCD codes for short, are codeswhose intersections with their Hermitian dual codes are trivial. The concept of LCD codes was inventedby Massey [7] in 1992. LCD codes have been applied in data storage, communication systems andcryptography. For example, it is known that LCD codes can be used against side-channel attacks andKeita Ishizuka; Research Center for Pure and Applied Mathematics Graduate School of Information Sciences,Tohoku University, Sendai 980–8579, Japan (email: keita.ishizuka.p5@dc.tohoku.ac.jp).229K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236fault injection attacks [3]. We note that a code C is a Hermitian LCD code if and only if its generatormatrix G satisfies detGGT 6= 0 [5].Two codes C,C ′ are equivalent if one can be obtained from the other by a permutation of thecoordinates and a multiplication of any coordinate by a nonzero scalar. We denote the equivalence oftwo codes C,C ′ by C ' C ′. Let G,G′ be generator matrices of two codes C,C ′ respectively. It is knownthat C ' C ′ if and only if G can be obtained from G′ by an elementary row operation, a permutation ofthe 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 [n, 2, d]codes is given as follows:d ≤{⌊4n5⌋if n ≡ 1, 2, 3 (mod 5),⌊4n5⌋− 1 otherwise.(1)Also, it was proved that for all n 6= 1, there exists a Hermitian LCD [n, 2, d] code which meets this upperbound. We say that a Hermitian LCD [n, 2, d] code is optimal if it meets this upper bound. It wasshown in [4] that any code over Fq2 is equivalent to some Hermitian LCD code for q ≥ 3. Furthermore,it was proved in [6] that a Hermitian LCD code leads to a construction of a maximal-entanglemententanglement-assisted quantum error correcting code. Motivated by the results, we are concerned withthe complete classification of optimal Hermitian LCD codes of dimension 2.This paper is organized as follows. In Section 2, we present a method to construct optimal HermitianLCD codes of dimension 2, including all inequivalent codes. Also, a method to classify optimal HermitianLCD codes of dimension 2 is given. In Section 3, we classify optimal Hermitian LCD codes of dimension2. Up to equivalence, the complete classification of optimal Hermitian LCD codes of dimension 2 isgiven. It is shown that all inequivalent codes have distinct weight enumerators, which is used for theclassification.2. Classification methodLet 0n be the zero vector of length n and 1n be the all-ones vector of length n. Let(a0, a1, a2, a3, a4, a5) be a tuple of nonnegative integers. We introduce the following notation:G(a0, a1, a2, a3, a4, a5) :=(1 0 0a0 0a1 1a2 1a3 1a4 1a50 1 0a0 1a1 0a2 1a3 ω1a4 ω21a5).We denote by C(a0, a1, a2, a3, a4, a5) the code whose generator matrix is G(a0, a1, a2, a3, a4, a5). By thesame argument as in [1], we obtain the following lemma.Lemma 2.1. Given a code C, define C∗ := {(x, 0) | x ∈ C}. Let C∗n,k denote the set of all inequivalentHermitian LCD [n, k] codes C such that the minimum weight of C⊥h is 1. Then there exists a set Cn−1,kof all inequivalent Hermitian LCD [n− 1, k] codes such that C∗n,k = {C∗ | C ∈ Cn−1,k}.We assume a0 = 0 by Lemma 2.1 and omit a0. Furthermore, throughout this paper, we use thefollowing notations:G(a) := G(a1, a2, a3, a4, a5), C(a) := C(a1, a2, a3, a4, a5), (2)respectively, to save space.Proposition 2.2. Let C be an [n, 2, d] code. Then there exist nonnegative integers a1, a2, a3, a4, a5 suchthat C ' C(a) and 1 + a2 + a3 + a4 + a5 = d.Proof. Let G be a generator matrix of the code C. By multiplying rows by some non-zero scalars, Gis changed to a generator matrix which consists only of the columns of G(a). Permuting the columns,230K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236G(a) is obtained from G. Hence it holds that C ' C(a). Since the minimum weight of C is d, we mayassume that the first row of G is a codeword with weight d, which yields 1 + a2 + a3 + a4 + a5 = d.Given a code C(a), we may assume without loss of generality that G(a) satisfies1 + a2 + a3 + a4 + a5 = d, (3)by Proposition 2.2. This assumption on a generator matrix reduces computations later.Lemma 2.3. Let C be an [n, 2, d] code C(a). Then C is a Hermitian LCD code if and only if C satisfiesthe following conditions:a1 = n− d− 1, (4)a2 ≤ n− d− 1, (5)a3, a4, a5 ≤ n− d, (6){a3 + a4 + a5 + a3a4 + a4a5 + a5a3 6≡ 0 (mod 2) if d is even,a3a4 + a4a5 + a5a3 6≡ n− d (mod 2) otherwise.(7)Proof. Suppose C is a Hermitian LCD [n, 2, d] code. Let G be a generator matrix of the code C. Letr1, r2 be the first and second rows of G respectively. The number of columns of G equals to the lengthn. Thus, it holds that2 + a1 + a2 + a3 + a4 + a5 = n. (8)Since the minimum weight of C is d, the following holds: wt(r2) ≥ d, wt(r1 + r2) ≥ d, wt(r1 + ωr2) ≥d, wt(r1 + ω2r2) ≥ d. By (3), we have wt(r1) = d. Substituting (8) in each equation, we obtain (4)through (6).The code C is a Hermitian LCD code if and only ifdetGGT= (r1, r1)h(r2, r2)h − (r2, r1)h(r1, r2)h 6= 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 n, d, a3, a4, a5 as elements of F4. Therefore, (4) through (7) hold if C is a Hermitian LCD[n, 2, d] code and vice versa.Given an [n, 2, d] code C(a), we define the following:bi := (n− d)− ai for 1 ≤ i ≤ 5. (9)Lemma 2.4. Let C be an [n, 2, d] code C(a). Then C is a Hermitian LCD code if and only if C satisfiesthe following conditions:b1 = 1, (10)b2 ≥ 1, (11)b3, b4, b5 ≥ 0,{b3 + b4 + b5 + b3b4 + b4b5 + b5b3 6≡ 0 (mod 2) if d is even,b3b4 + b4b5 + b5b3 6≡ 0 (mod 2) otherwise.231K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236Proof. The result follows from Lemma 2.3.Given an [n, 2, d] code C(a), we define the following:∆ := 4n− 5d. (12)Lemma 2.5. Let C be a Hermitian LCD [n, 2, d] code. Then C is optimal if and only if the value of ∆with respect to n is given as follows:∆ =5 if n ≡ 0 (mod 5),4 if n ≡ 1 (mod 5),3 if n ≡ 2 (mod 5),2 if n ≡ 3 (mod 5),6 if n ≡ 4 (mod 5).Proof. The result follows from (12).Lemma 2.6. Let C be a code C(a). If C is a Hermitian LCD code, then it holds that0 ≤ b3, b4, b5 ≤ ∆.Proof. Substituting b2, b3, b4, b5,∆ in (3), we obtainb2 = ∆ + 1− (b3 + b4 + b5). (13)Combining with (11), we obtain b3+b4+b5 ≤ ∆. Since 0 ≤ b3, b4, b5, it follows that 0 ≤ b3, b4, b5 ≤ ∆.Lemma 2.7.C(a) ' C(a1, a2, a5, a3, a4). (14)Proof. Multiply the second row of G(a) by ω. Permuting the columns, the result follows. Note thatwe may assume that the nonzero entry of a column is 1, provided that the entry of the other column is0.By Lemma 2.7, we may assume a3 ≥ a4, a5. Notice that a3 ≥ a4, a5 if and only if b3 ≤ b4, b5 by (9).3. Optimal Hermitian LCD codes of dimension 2By Lemmas 2.4 through 2.7, it suffices to calculate all b3, b4, b5 satisfying0 ≤ b3 ≤ b4, b5 ≤ ∆, (15){b3 + b4 + b5 + b3b4 + b4b5 + b5b3 6= 0 if d is even,b3b4 + b4b5 + b5b3 6= 0 otherwise,(16)in order to obtain optimal Hermitian LCD codes of dimension 2, including all inequivalent codes. Noticethat b1, b2 are obtained by (10), (13) respectively. Our computer search found all integers b3, b4, b5 satisfy-ing (15) and (16). This calculation was done by Magma [2]. Recall that a1, a2, a3, a4, a5 are obtained fromb1, b2, b3, b4, b5 by (9). For optimal Hermitian LCD codes of dimension 2, the integers a1, a2, a3, a4, a5 arelisted in Table 1, where the rows are in lexicographical order with respect to a1, a2, a3, a4, a5, and m is anonnegative integer.232K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236Table 1: Optimal Hermitian LCD codes of dimension 2n Code (a1, a2, a3, a4, a5) mn = 5m C5m,1 (m,m,m,m,m− 2) m ≥ 2C5m,2 (m,m,m,m− 2,m) m ≥ 2C5m,3 (m,m− 1,m+ 1,m,m− 2) m ≥ 2C5m,4 (m,m− 1,m+ 1,m− 2,m) m ≥ 2C5m,5 (m,m− 1,m,m,m− 1) m ≥ 1C5m,6 (m,m− 1,m,m− 1,m) m ≥ 1C5m,7 (m,m− 2,m,m,m) m ≥ 2C5m,8 (m,m− 3,m+ 1,m,m) m ≥ 3n = 5m+ 1 C5m+1,1 (m,m,m+ 1,m,m− 2) m ≥ 2C5m+1,2 (m,m,m+ 1,m− 2,m) m ≥ 2C5m+1,3 (m,m,m,m,m− 1) m ≥ 1C5m+1,4 (m,m,m,m− 1,m) m ≥ 1C5m+1,5 (m,m− 1,m+ 1,m+ 1,m− 2) m ≥ 2C5m+1,6 (m,m− 1,m+ 1,m,m− 1) m ≥ 1C5m+1,7 (m,m− 1,m+ 1,m− 1,m) m ≥ 1C5m+1,8 (m,m− 1,m+ 1,m− 2,m+ 1) m ≥ 2C5m+1,9 (m,m− 2,m+ 1,m,m) m ≥ 2C5m+1,10 (m,m− 3,m+ 1,m+ 1,m) m ≥ 3C5m+1,11 (m,m− 3,m+ 1,m,m+ 1) m ≥ 3n = 5m+ 2 C5m+2,1 (m,m,m,m,m) m ≥ 0C5m+2,2 (m,m− 1,m+ 1,m,m) m ≥ 1n = 5m+ 3 C5m+3,1 (m,m− 1,m+ 1,m+ 1,m) m ≥ 1C5m+3,2 (m,m,m+ 1,m,m) m ≥ 0C5m+3,3 (m,m− 1,m+ 1,m,m+ 1) m ≥ 1n = 5m+ 4 C5m+4,1 (m+ 1,m+ 1,m+ 1,m+ 1,m− 2) m ≥ 2C5m+4,2 (m+ 1,m+ 1,m+ 1,m,m− 1) m ≥ 1C5m+4,3 (m+ 1,m+ 1,m+ 1,m− 1,m) m ≥ 1C5m+4,4 (m+ 1,m+ 1,m+ 1,m− 2,m+ 1) m ≥ 2C5m+4,5 (m+ 1,m+ 1,m+ 2,m+ 1,m− 3) m ≥ 3C5m+4,6 (m+ 1,m+ 1,m+ 2,m− 1,m− 1) m ≥ 1C5m+4,7 (m+ 1,m+ 1,m+ 2,m− 3,m+ 1) m ≥ 3C5m+4,8 (m+ 1,m,m+ 1,m,m) m ≥ 0C5m+4,9 (m+ 1,m,m+ 2,m+ 1,m− 2) m ≥ 2C5m+4,10 (m+ 1,m,m+ 2,m+ 2,m− 3) m ≥ 3C5m+4,11 (m+ 1,m,m+ 2,m,m− 1) m ≥ 1C5m+4,12 (m+ 1,m,m+ 2,m− 1,m) m ≥ 1C5m+4,13 (m+ 1,m,m+ 2,m− 2,m+ 1) m ≥ 2C5m+4,14 (m+ 1,m,m+ 2,m− 3,m+ 2) m ≥ 3C5m+4,15 (m+ 1,m− 1,m+ 1,m+ 1,m) m ≥ 1C5m+4,16 (m+ 1,m− 1,m+ 1,m,m+ 1) m ≥ 1C5m+4,17 (m+ 1,m− 1,m+ 2,m+ 1,m− 1) m ≥ 1C5m+4,18 (m+ 1,m− 1,m+ 2,m− 1,m+ 1) m ≥ 1C5m+4,19 (m+ 1,m− 2,m+ 2,m+ 1,m) m ≥ 2C5m+4,20 (m+ 1,m− 2,m+ 2,m+ 2,m− 1) m ≥ 2C5m+4,21 (m+ 1,m− 2,m+ 2,m,m+ 1) m ≥ 2C5m+4,22 (m+ 1,m− 2,m+ 2,m− 1,m+ 2) m ≥ 2C5m+4,23 (m+ 1,m− 3,m+ 2,m+ 1,m+ 1) m ≥ 3C5m+4,24 (m+ 1,m− 4,m+ 2,m+ 1,m+ 2) m ≥ 4C5m+4,25 (m+ 1,m− 4,m+ 2,m+ 2,m+ 1) m ≥ 4Lemma 3.1. Suppose a3 is a positive integer. ThenC(a) ' C(a1, a3 − 1, a2 + 1, a5, a4). (17)Proof. Add the second row of G(a) to the first row. Permuting the columns, the result follows. Recall thatC(a) is defined in (2).233K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236Table 2. Equivalent optimal Hermitian LCD codes of dimension 2n Coden = 5m C5m,7 '17,14 C5m,6 '14 C5m,5C5m,8 '17,14,17 C5m,3 '17 C5m,2 '14 C5m,1 '17 C5m,4n = 5m+ 1 C5m+1,9 '17,14,17 C5m+1,6 '17 C5m+1,4 '14 C5m+1,3 '17 C5m+1,7C5m+1,11 '17,14,17 C5m+1,5 '17,14 C5m+1,1 '18 C5m+1,2'14,17 C5m+1,8 '17,14,17 C5m+1,10n = 5m+ 2 C5m+2,1 '17 C5m+2,2n = 5m+ 3 C5m+3,1 '17,14 C5m+3,2 '14,17 C5m+3,3n = 5m+ 4 C5m+4,8 '14,17 C5m+4,16 '14 C5m+4,15C5m+4,6 '14,17 C5m+4,22 '14 C5m+4,20C5m+4,21 '17,14,17 C5m+4,17 '17,14,17 C5m+4,12'17 C5m+4,2 '18 C5m+4,3 '17 C5m+4,11'17,14,14,17 C5m+4,19 '17,14,14,17 C5m+4,18C5m+4,23 '17,14,17 C5m+4,9'17 C5m+4,4 '18 C5m+4,1 '17 C5m+4,13C5m+4,24 '17,14,17 C5m+4,10 '17,14 C5m+4,5 '18 C5m+4,7'14,17 C5m+4,14 '17,14,17 C5m+4,25Lemma 3.2. Suppose 1 + a1 + a3 + a4 + a5 = d. ThenC(a) ' C(a2, a1, a3, a5, a4). (18)Proof. Interchange the first row and the second row of G(a). Permuting the columns, the result follows.By Lemmas 2.7 through 3.2, we have the equivalences among some codes listed in Table 1, which are displayedin Table 2. Note that C 'i C′ denotes the two codes C,C′ are equivalent by (i). Also, C 'i,j C′ denotes that,given two codes C,C′, there exists a code C′′ such that C 'i C′′ 'j C′.Table 3 gives the weight enumerators of representatives in Table 2. The weight enumerator is given by1 + 3ywt(r1) + 3ywt(r2) + 3ywt(r1+r2) + 3ywt(r1+ωr2) + 3ywt(r1+ω2r2), where r1, r2 be the first and second rows ofG(a) respectively. Since the weight enumerators are distinct, the codes in Table 3 are inequivalent. Table 4 givesthe classification of optimal Hermitian LCD codes of dimension 2, with the case where n is so small that somecodes in Table 1 do not exist.Recall that we have assumed a0 = 0 by Lemma 2.1. It follows from (1) that there exists an optimal HermitianLCD [n, 2] code C such that the minimum weight of C⊥h equals to 1 if and only if n ≡ 4 (mod 5). Consequently,we obtain the following theorem.Theorem 3.3. (i) Up to equivalence, there exist two optimal Hermitian LCD [5m, 2, 4m− 1] codes for everyinteger m with m ≥ 2.(ii) Up to equivalence, there exist two optimal Hermitian LCD [5m + 1, 2, 4m] codes for every integer m withm ≥ 2.(iii) Up to equivalence, there exists a unique optimal Hermitian LCD [5m+ 2, 2, 4m+ 1] code for every integerm with m ≥ 0.(iv) Up to equivalence, there exists a unique optimal Hermitian LCD [5m+ 3, 2, 4m+ 2] code for every integerm with m ≥ 0.(v) Up to equivalence, there exist six optimal Hermitian LCD [5m+4, 2, 4m+2] codes for every integer m withm ≥ 3. One of them is the code such that the minimum weight of the Hermitian dual code is 1.234K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236Table 3. Weight enumerators of representativesn Code Weight Enumeratorn = 5m C5m,7 1 + 3y4m−1 + 9y4m + 3y4m+1C5m,8 1 + 6y4m−1 + 6y4m + 3y4m+2n = 5m+ 1 C5m+1,9 1 + 6y4m + 6y4m+1 + 3y4m+2C5m+1,1 1 + 9y4m + 3y4m+1 + 3y4m+3n = 5m+ 2 C5m+2,1 1 + 6y4m+1 + 6y4m+2 + 3y4m+3n = 5m+ 3 C5m+3,1 1 + 9y4m+2 + 6y4m+3n = 5m+ 4 C5m+4,8 1 + 3y4m+2 + 6y4m+3 + 6y4m+4C5m+4,6 1 + 9y4m+2 + 6y4m+5C5m+4,21 1 + 6y4m+2 + 3y4m+3 + 3y4m+4 + 3y4m+5C5m+4,23 1 + 6y4m+2 + 6y4m+3 + 3y4m+6C5m+4,24 1 + 9y4m+2 + 3y4m+3 + 3y4m+7Table 4. Classification of optimal Hermitian LCD codes of dimension 2n m Coden = 5m m = 1 C5m,7m ≥ 2 C5m,7, C5m,8n = 5m+ 1 m = 1 C5m+1,9m ≥ 2 C5m+1,9, C5m,1n = 5m+ 2 m ≥ 0 C5m+2,1n = 5m+ 3 m ≥ 0 C5m+3,1n = 5m+ 4 m = 0 C5m+4,8m = 1 C5m+4,8, C5m+4,6, C5m+4,21m = 2 C5m+4,8, C5m+4,6, C5m+4,21, C5m+4,23m ≥ 3 C5m+4,8, C5m+4,6, C5m+4,21, C5m+4,23, C5m+4,244. Concluding remarksA natural extension of this work is to classify the quaternary Hermitian LCD codes of larger dimensions. Inthe case where the dimension is 3, there are 64 different codewords. By a method similar to that in Proposition 2.2,the number of column vectors we need consider is reduced to 22. However, this is still a large number. Therefore,it is difficult to extend our method to classify Hermitian LCD codes of larger dimensions.Acknowledgment: The author would like to thank supervisor Professor Masaaki Harada for introducingthe problem, useful discussions and his encouragement. Also, the author would like to thank the reviewers fortheir thoughtful comments.235K. Ishizuka / J. Algebra Comb. Discrete Appl. 7(3) (2020) 229–236References[1] M. Araya, M. Harada, On the classification of linear complementary dual codes, Discrete Math. 342(2019) 270–278.[2] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. SymbolicComput. 24 (1997) 235–265.[3] C. Carlet, S. Guilley, Complementary dual codes for counter–measures to side–channel attacks, Adv.Math. Commun. 10 (2016) 131–150.[4] C. Carlet, S. Mesnager, C. Tang, Y. Qi, R. Pellikaan, Linear codes over Fq are equivalent to LCDcodes for q > 3, IEEE Trans. Inform. Theory 64 (2018) 3010–3017.[5] C. Güneri, B. Özkaya, P. Solé, Quasi–cyclic complementary dual codes, Finite Fields Appl. 42 (2016)67–80.[6] L. Lu, R. Li, L. Guo, Q. Fu, Maximal entanglement entanglement–assisted quantum codes constructedfrom linear codes, Quantum Inf. Process. 14 (2015) 165–182.[7] J. L. Massey, Linear codes with complementary duals, Discrete Math. 106/107 (1992) 337–342.236

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