Hermitian Self-orthogonal Codes Research Articles

Several classes of Galois self-orthogonal MDS codes and related applications

Let q=ph be a prime power and e be an integer with 0≤e≤h−1. e-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e=0) and Hermitian self-orthogonal codes (e=h2 and h is even). In this paper, we propose two general methods to construct e-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e-Galois self-orthogonal (extended) GRS codes with odd q and 2e|h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e′-Galois self-orthogonal maximum distance separable (MDS) codes for all possible e′ satisfying 0≤e′≤h−1, new e-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e-Galois hull via the punctured codes are derived. Moreover, some new q-ary quantum MDS codes with length greater than q+1 and minimum distance greater than q2+1 are obtained.

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

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q^2-ary linear codes. This result can be generalized to q^{2 m}-ary linear codes, 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, improving others in the literature.

Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applications

<p style='text-indent:20px;'>Hermitian self-orthogonal codes form an essential ingredient in the construction of quantum stabilizer codes. In this paper, we present two new classes of Hermitian self-orthogonal MDS codes from twisted generalized Reed-Solomon codes. The constructed codes are monomially inequivalent to generalized Reed-Solomon codes. Two new classes of Hermitian LCD MDS codes can then be derived. Finally, based on the constructed Hermitian self-orthogonal MDS codes, we present two classes of quantum MDS codes with new parameters.</p>

The geometry of Hermitian self-orthogonal codes

We prove that if n >k^2 then a k-dimensional linear code of length n over {mathbb F}_{q^2} has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than n common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.

On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes

Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ </tex-math></inline-formula> be the primitive BCH code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{r}$ </tex-math></inline-formula> of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r^{m}-1$ </tex-math></inline-formula> with designed distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{r}$ </tex-math></inline-formula> is the finite field of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> . In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ </tex-math></inline-formula> and its hull in both Hermitian and Euclidean cases for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \le \delta \le r^{2}-1$ </tex-math></inline-formula> . We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.

Hermitian Rank Metric Codes and Duality

In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for $\mathbb {F}_{q^{2}}$ -linear matrix codes in the ambient space $(\mathbb {F}_{q^{2}})_{n,m}$ and for both $\mathbb {F}_{q^{2}}$ -additive codes and $\mathbb {F}_{q^{2m}}$ -linear codes in the ambient space $\mathbb {F}_{q^{2m}}^{n}$ . Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of $q^{m}$ -duality between bases of $\mathbb {F}_{q^{2m}}$ over $\mathbb {F}_{q^{2}}$ and prove that a $q^{m}$ -self dual basis exists if and only if $m$ is an odd integer. We obtain connections on the dual codes in $\mathbb {F}_{q^{2m}}^{n}$ and $(\mathbb {F}_{q^{2}})_{n,m}$ with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in $\mathbb {F}_{q^{2m}}^{n}$ and $(\mathbb {F}_{q^{2}})_{n,m}$ . Furthermore, we present connections between Hermitian $\mathbb {F}_{q^{2}}$ -additive codes and Euclidean $\mathbb {F}_{q^{2}}$ -additive codes in $\mathbb {F}_{q^{2m}}^{n}$ .

Self-orthogonal Codes over Fq + uFq and Fq + uFq + u 2Fq

In this paper, we establish a mass formula for Euclidean and Hermitian self-orthogonal codes over the finite ring Fq + uFq, where Fq is the finite field of order q and u2 = 0. We also establish a mass formula for Euclidean self-orthogonal codes over the finite ring Fq + uFq + u2Fq, with u3 = 0 and characteristic of Fq is odd. These mass formulas are used to give a classification of Euclidean and Hermitian self-orthogonal codes over F2 + uF2 and F3 + uF3 of small lengths.

The images of constacyclic codes and new quantum codes

Let q be a prime power and $$m\ge 2$$ be a positive integer. A sufficient condition for the $$q^2$$-ary images of constacyclic codes over $${\mathbb {F}}_{q^{2m}}$$ to be Hermitian self-orthogonal is presented. Hermitian self-orthogonal codes over $${\mathbb {F}}_{q^{2}}$$ are obtained as the images of constacyclic codes over $${\mathbb {F}}_{q^{2m}}$$. Two classes of quantum codes are derived by employing the Hermitian construction. The construction produces quantum codes with better parameters than the previously known ones.

Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices

We define the notion of an orbit matrix with respect to standard weighing matrices, and with respect to types of weighing matrices with entries in a finite field. In the latter case we primarily restrict our attention the fields of order 2, 3 and 4. We construct self-orthogonal and Hermitian self-orthogonal linear codes over finite fields from these types of weighing matrices and their orbit matrices respectively. We demonstrate that this approach applies to several combinatorial structures such as Hadamard matrices and balanced generalized weighing matrices. As a case study we construct self-orthogonal codes from some weighing matrices belonging to some well known infinite families, such as the Paley conference matrices, and weighing matrices constructed from ternary periodic Golay pairs.

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

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.

Construction of some new quantum MDS codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, we mainly apply a new method of classical Hermitian self-orthogonal codes to construct three classes of new quantum MDS codes, and these quantum MDS codes provide large minimum distance.

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).

New q-ary quantum MDS codes with distances bigger than $$\frac{q}{2}$$ q 2

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474---1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS $$[[n,n-2d+2,d]]_q$$[[n,n-2d+2,d]]q codes with minimum distances $$d>\frac{q}{2}$$d>q2 for sparse lengths $$n>q+1$$n>q+1. In the case $$n=\frac{q^2-1}{m}$$n=q2-1m where $$m|q+1$$m|q+1 or $$m|q-1$$m|q-1 there are complete results. In the case $$n=\frac{q^2-1}{m}$$n=q2-1m while $$m|q^2-1$$m|q2-1 is neither a factor of $$q-1$$q-1 nor $$q+1$$q+1, no q-ary quantum MDS code with $$d> \frac{q}{2}$$d>q2 has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over $$\mathbf{F}_{q^2}$$Fq2. Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form $$\frac{w(q^2-1)}{u}$$w(q2-1)u and minimum distances $$d > \frac{q}{2}$$d>q2 are presented.

A Construction of New Quantum MDS Codes

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 = q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + 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 (q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + 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 F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> of a system of homogenous equations over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .

QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER $\mathbb{F}_4 + u\mathbb{F}_4$

We give a construction for quantum codes from linear and cyclic codes over [Formula: see text]. We derive Hermitian self-orthogonal codes over [Formula: see text] as Gray images of linear and cyclic codes over [Formula: see text]. In particular, we use two binary codes associated with a cyclic code over [Formula: see text] of odd length to determine the parameters of the corresponding quantum code.

TERNARY QUANTUM CODES OF MINIMUM DISTANCE THREE

Based on finite field and combinatorics theory, an elementary recursive construction is used to construct Hermitian self-orthogonal codes over 𝔽9, the Galois field with nine elements, with dual distance three for all n ≥ 4. Consequently, good ternary linear quantum codes of minimum distance three for such length n are obtained. These ternary linear quantum codes are optimal or near optimal according to the quantum Singleton bound and quantum Hamming bound.

Generalized weighing matrices and self-orthogonal codes

It is shown that the row spaces of certain generalized weighing matrices, Bhaskar Rao designs, and generalized Hadamard matrices over a finite field of order q are hermitian self-orthogonal codes. Certain matrices of minimum rank yield optimal codes. In the special case when q = 4 , the codes are linked to quantum error-correcting codes, including some codes with optimal parameters.