arXiv:2012.11998 · Designs Codes and Cryptography

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Carlos Galindo, Fernando Hernando

Published Mar 21, 2022• 8 citations

Abstract

Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal $$q^2$$ q 2 -ary linear codes. This result can be generalized to $$q^{2 m}$$ q 2 m -ary linear codes, $$m > 1$$ m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with $$q \ne 2$$ q ≠ 2 , improving others in the literature.

DOI: 10.1007/s10623-022-01018-2

Related papers

New quantum codes from homothetic-BCH codes

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz

Abstract We introduce homothetic-BCH codes. These are a family of $$q^2$$ q 2 -ary classical codes $$\mathcal {C}$$ C of length $$\lambda n_1$$ λ n 1 , where $$\lambda $$ λ and $$n_1$$ n 1 are suitable positive integers such that the punctured code $$\mathcal {B}$$ B of $$\mathcal {C}$$ C in the last $$\lambda n_1 -n_1$$ λ n 1 - n 1 coordinates is a narrow-sense BCH code of length $$n_1$$ n 1 . We prove that whenever $$\mathcal {B}$$ B is Hermitian self-orthogonal, so is $$\mathcal {C}$$ C . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

arXiv:2503.13069Computational and Applied MathematicsDec 18, 20251 citation

Quasi-cyclic Hermitian construction of binary quantum codes

Liangdong Lü, Zhang, Gaochi, Feng, Ganyu +1 more

In this paper, we propose a sufficient condition for a family of 2-generator self-orthogonal quasi-cyclic codes with respect to Hermitian inner product. Supported in the Hermitian construction, we show algebraic constructions of good quantum codes. 30 new binary quantum codes with good parameters improving the best-known lower bounds on minimum distance in Grassl's code tables \cite{Grassl:codetables} are constructed.

arXiv:2210.04487Oct 10, 20220 citations

A Construction of New Quantum MDS Codes

Lingfei Jin, Chaoping Xing

arXiv:1311.3009IEEE Transactions on Information TheoryJan 31, 2014108 citations

Several Families of Self-Orthogonal Codes and Their Applications in Optimal Quantum Codes and LCD Codes

Xinran Wang, Ziling Heng

IEEE Transactions on Information TheoryNov 13, 202316 citations

Nonbinary quantum stabilizer codes

Alexei Ashikhmin, Emanuel H. Knill

We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between self-orthogonal codes over F/sub 4/ and binary quantum codes to one between self-orthogonal codes over F(q/sup 2/) and q-ary quantum codes for any prime power q.

arXiv:quant-ph/0005008IEEE Transactions on Information TheoryJan 1, 2001627 citations

Quantum stabilizer codes construction from Hermitian self-orthogonal codes over GF(4)

Duc Manh Nguyen, Sunghwan Kim

Journal of Communications and NetworksJun 1, 201818 citations

arXiv:2012.11998 · Designs Codes and Cryptography

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Carlos Galindo, Fernando Hernando

Published Mar 21, 2022• 8 citations

arXiv HTML PDF

Abstract

Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal

q^2

q 2 -ary linear codes. This result can be generalized to

q^{2 m}

q 2 m -ary linear codes,

m > 1

m > 1 . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de , 2020) and new q-ary ones, with

q \ne 2

q ≠ 2 , improving others in the literature.

DOI: 10.1007/s10623-022-01018-2

Related papers

New quantum codes from homothetic-BCH codes

Carlos Galindo, Fernando Hernando, Helena Martín-Cruz

Abstract We introduce homothetic-BCH codes. These are a family of $q^2$ q 2 -ary classical codes $\mathcal {C}$ C of length $\lambda n_1$ λ n 1 , where $\lambda$ λ and $n_1$ n 1 are suitable positive integers such that the punctured code $\mathcal {B}$ B of $\mathcal {C}$ C in the last $\lambda n_1 -n_1$ λ n 1 - n 1 coordinates is a narrow-sense BCH code of length $n_1$ n 1 . We prove that whenever $\mathcal {B}$ B is Hermitian self-orthogonal, so is $\mathcal {C}$ C . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.Abstract We introduce homothetic-BCH codes. These are a family of $$q^2$$ q 2 -ary classical codes $$\mathcal {C}$$ C of length $$\lambda n_1$$ λ n 1 , where $$\lambda $$ λ and $$n_1$$ n 1 are suitable positive integers such that the punctured code $$\mathcal {B}$$ B of $$\mathcal {C}$$ C in the last $$\lambda n_1 -n_1$$ λ n 1 - n 1 coordinates is a narrow-sense BCH code of length $$n_1$$ n 1 . We prove that whenever $$\mathcal {B}$$ B is Hermitian self-orthogonal, so is $$\mathcal {C}$$ C . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

arXiv:2503.13069Computational and Applied MathematicsDec 18, 20251 citation

Quasi-cyclic Hermitian construction of binary quantum codes

Liangdong Lü, Zhang, Gaochi, Feng, Ganyu +1 more

arXiv:2210.04487Oct 10, 20220 citations

A Construction of New Quantum MDS Codes

Lingfei Jin, Chaoping Xing

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.

arXiv:1311.3009IEEE Transactions on Information TheoryJan 31, 2014108 citations

Several Families of Self-Orthogonal Codes and Their Applications in Optimal Quantum Codes and LCD Codes

Xinran Wang, Ziling Heng

Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime powerq, it is in general difficult to constructq-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a generalq-ary linear code is self-orthogonal or not. In this paper, we mainly present several families ofq-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families ofq-ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families ofq-ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families ofq-ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.Self-orthogonal codes have nice applications in many areas including quantum codes, lattices and LCD codes. For a prime powerq, it is in general difficult to constructq-ary self-orthogonal codes. In the literature, there exists no simple method to judge whether a generalq-ary linear code is self-orthogonal or not. In this paper, we mainly present several families ofq-ary self-orthogonal codes and study their applications in quantum codes and LCD codes. Firstly, several families ofq-ary linear codes are constructed by some special defining sets. These codes are proved to be self-orthogonal. To this end, we determine the numbers of solutions of some systems of equations over finite fields. Secondly, three families ofq-ary quantum codes with unbounded length and minimum distance three are constructed from the self-orthogonal codes. These quantum codes are optimal according to the quantum Hamming bound. In particular, some of them have better parameters than known ones. Thirdly, several families ofq-ary LCD codes are constructed from the self-orthogonal codes. Many optimal or almost optimal binary and ternary LCD codes are produced by our constructions. Some binary and ternary LCD codes have better parameters than known ones.

IEEE Transactions on Information TheoryNov 13, 202316 citations

Nonbinary quantum stabilizer codes

Alexei Ashikhmin, Emanuel H. Knill

arXiv:quant-ph/0005008IEEE Transactions on Information TheoryJan 1, 2001627 citations

Quantum stabilizer codes construction from Hermitian self-orthogonal codes over GF(4)

Duc Manh Nguyen, Sunghwan Kim

In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.

Journal of Communications and NetworksJun 1, 201818 citations

Abstract Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal

q^2

q 2 -ary linear codes. This result can be generalized to

q^{2 m}

q 2 m -ary linear codes,

m > 1

q \ne 2

q ≠ 2 , improving others in the literature.

Abstract We introduce homothetic-BCH codes. These are a family of

q^2

q 2 -ary classical codes

\mathcal {C}

C of length

\lambda n_1

λ n 1 , where

\lambda

λ and

n_1

n 1 are suitable positive integers such that the punctured code

\mathcal {B}

B of

\mathcal {C}

C in the last

\lambda n_1 -n_1

λ n 1 - n 1 coordinates is a narrow-sense BCH code of length

n_1

n 1 . We prove that whenever

\mathcal {B}

B is Hermitian self-orthogonal, so is

\mathcal {C}

C . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.