© Wuhan University 2023

0 Introduction

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. They generalize all major quantum error protection schemes, and therefore are exceptionally versatile.

The subsystem codes are constructions of quantum codes combining the features of decoherence-free subspaces, noiseless subsystems, and quantum error-correcting codes. These codes can potentially provide attractive features, including streamlined syndrome calculation and a diverse range of easily implementable fault-tolerant operations. An $({(n,K,R,d))}_q$ subsystem code is a KR-dimensional subspace $Q$ of $C^{q^n}$, which is decomposed into a tensor product $Q=A\otimes B$ of a K-dimensional vector space $A$ and an R-dimensional vector space $B$ such that all errors of weight less than $d$ can be detected by $A$. The vector spaces $A$ and $B$ are respectively referred to as the subsystem $A$ and the co-subsystem $B$. We also use bracket notation $[{[n,k,r,d]]}_q$ to write the parameters of an $((n,q^k,q^r{,d))}_q$ subsystem code in more straightforward form. For some background on subsystem codes, see the next section.

Aly et al^[1-4] gave various methods to derive subsystem codes from classical codes over binary and non-binary fields and presented subsystem codes' families. In Ref.[5], Leng and Ma provided two methods to construct good non-binary subsystem codes. The first one is derived from quantum codes applied to non-narrow-sense BCH codes. The second one is derived from the technique of defining sets of classical cyclic codes. Recently, Qian and Zhang ^[6,7] constructed two new classes of subsystem maximum distance separable (MDS) codes using two classes of classical cyclic codes.

Inspired by these works, in this paper, we construct three classes of Clifford subsystem MDS codes based on linear codes, which have parameters as follows:

(i) $[{[n,n-\mathrm{2}k+\mathrm{2}a-\mathrm{1,2}a-\mathrm{1},k-\mathrm{2}a+\mathrm{2}]]}_q$, where $n=q-\mathrm{1},a\ge \mathrm{1}$, and $\mathrm{2}a-\mathrm{1}\le k<\lceil \frac{n+\mathrm{2}a-\mathrm{1}}{\mathrm{2}}\rceil$.

(ii) $[{[n,n-\mathrm{2}k+\mathrm{1,1},k]]}_q$, where $n|q-\mathrm{1}$, $\mathrm{0}\le k<\lceil \frac{n+\mathrm{1}}{\mathrm{2}}\rceil$.

(iii) $[{[q+\mathrm{1},q-\mathrm{2}k+\mathrm{2,1},k]]}_q$, where $\mathrm{1}\le k<\lceil \frac{q+\mathrm{1}}{\mathrm{2}}\rceil$.

We conclude this introduction with a description of each section in this paper. Section 1 revisits the fundamentals and results of linear codes and subsystem codes. In Section 2 details, we construct three classes of Clifford subsystem MDS codes using Reed-Solomon codes and extended generalized Reed-Solomon codes over $F_q$. In Section 3, we compare the Clifford subsystem codes and give a summary of this work.

1 Preliminaries

In this section, we first recall some basic concepts and results about linear codes and subsystem codes necessary for the development of this work. We refer to Refs.[1, 4, 8] for more details.

Throughout this paper, let $F_q$ be the finite field with $q=p^e$ elements, where $p$ is a prime number and $e\ge \mathrm{1}$ is an integer. For a positive integer $n$, let $F_q^n$ denote the vector space of all $n$-tuples over $F_q$. A linear ${[n,k]}_q$ code $C$ over $F_q$ is a $k$-dimensional subspace of $F_q^n$. The Hamming weight $\mathrm{w}\mathrm{t}(\mathit{c})$ of a codeword $\mathit{c}\in C$ is the number of nonzero components of $\mathit{c}$. The Hamming distance of two codewords ${\mathit{c}}_{\mathrm{1}},{\mathit{c}}_{\mathrm{2}}\in C$, is $d({\mathit{c}}_{\mathrm{1}},{\mathit{c}}_{\mathrm{2}})=\mathrm{w}\mathrm{t}({\mathit{c}}_{\mathrm{2}}-{\mathit{c}}_{\mathrm{1}})$. The minimum Hamming distance $d(C)$ of $C$ is the smallest Hamming distance between any two distinct codewords $C$. An ${[n,k,d]}_q$ code is an ${[n,k]}_q$ linear code with the minimum Hamming distance $d$.

Let $C^{q^n}=C^q\otimes \cdots \otimes C^q$. Let $|x〉$ be the vectors of an orthonormal basis of $C^q$, where the labels $x$ are elements of $F_q$. Then $C^{q^n}$ has the following orthonormal basis $\{|c〉=|c_{\mathrm{1}}{c}_{\mathrm{2}}\cdots c_n〉=|c_{\mathrm{1}}〉\otimes |c_{\mathrm{2}}〉\otimes \cdots \otimes |c_n〉:c$$=(c_{\mathrm{1}},c_{\mathrm{2}},\cdots ,c_n)\in F_q^n\}$.

If $S$ is a set, then $\left|S\right|$ denotes the cardinality of the set $S$. We use the notation $(\mathit{a}|\mathit{b})=(a_{\mathrm{1}},\cdots ,a_n|b_{\mathrm{1}},\cdots ,b_n)$ to denote concatenation of $\mathit{a}=(a_{\mathrm{1}},\cdots ,a_n),\mathit{b}=(b_{\mathrm{1}},\cdots ,b_n)\in F_q^n$.

The symplectic weight of $(\mathit{a}|\mathit{b})\in F_q^{\mathrm{2}n}$ is defined as $\mathrm{s}\mathrm{w}\mathrm{t}(\mathit{a}|\mathit{b})=\{(a_i,b_i)\ne (\mathrm{0,0}):\mathrm{1}\le i\le n\}$.

The trace-symplectic product of two vectors $\mathit{u}=(\mathit{a}|\mathit{b})$ and $\mathit{v}=({\mathit{a}}^{\text{'}}|{\mathit{b}}^{\text{'}})$ in $F_q^{\mathrm{2}n}$ is defined as

${〈\mathit{u}|\mathit{v}〉}_s=\mathrm{t}{\mathrm{r}}_{q/p}({\mathit{a}}^{\text{'}}\cdot \mathit{b}-\mathit{a}\cdot {\mathit{b}}^{\text{'}})$

where $\mathit{x}\cdot \mathit{y}$ denotes the dot product and $\mathrm{t}{\mathrm{r}}_{q/p}$ represents the trace from $F_q$ to the subfield $F_p$, i.e., $\mathrm{t}{\mathrm{r}}_{q/p}(a)=a+a^p+\cdots$$+a^{p^{e-\mathrm{1}}}$. The trace-symplectic dual of a code $C\in F_q^{\mathrm{2}n}$ is defined as $C^{{\perp }_s}=\{\mathit{u}\in F_q^{\mathrm{2}n}:{〈\mathit{u}|\mathit{v}〉}_s=\mathrm{0}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathit{v}\in C\}$.

Consider $a,b\in F_q$, the unitary linear operators $X(a)$ and $Z(b)$ in $C^q$ are defined by $X(a)|x〉=|x+a〉$ and $Z(b)|x〉=w^{\mathrm{t}{\mathrm{r}}_{q/p}(bx)}|x〉$, respectively, where $w=\mathrm{e}\mathrm{x}\mathrm{p}(\mathrm{2}\mathrm{\pi }i/p)$ is a primitive $p$-th root of unity.

Let $G_n=\{w^c{E}_{\mathrm{1}}\otimes \cdots \otimes E_n:E_i=X(a_i)Z(b_i),a_i,b_i\in$$F_q,c\in F_p\}$.

Then $G_n$ is called the error group on $C^{q^n}$.

The weight of an error $E=E_{\mathrm{1}}\otimes E_{\mathrm{2}}\otimes \cdots \otimes E_n$ in $G_n$ is defined as the number of $E_i$ which are not equal to identity, and it is denoted by $\mathrm{w}\mathrm{t}(E)$. We can also associate with $E$ a vector $\tilde{E}=(a_{\mathrm{1}},\cdots ,a_n|b_{\mathrm{1}},\cdots ,b_n)\in F_q^{\mathrm{2}n}$. We have

$\mathrm{w}\mathrm{t}(E)=\mathrm{s}\mathrm{w}\mathrm{t}(\tilde{E})=\left|\{(a_i,b_i)\ne (\mathrm{0,0})|\mathrm{1}\le i\le n\}\right|$

Every nontrivial normal subgroup $H$ in $G_n$ defines a subsystem code $Q$. Let $C_{G_n}(H)$ be the centralizer of $H$ in $G_n$ and $Z(H)$ the center of $H$. As a subspace, the subsystem code $Q$ defined by $H$ is precisely the same as the stabilizer code defined by $Z(H)$. By Ref.[9], Theorem 4, $Q$ can be decomposed as $A\otimes B$ where $\mathrm{d}\mathrm{i}\mathrm{m}B={\left|H:Z(H)\right|}^{\mathrm{1}/\mathrm{2}}$ and

$\mathrm{d}\mathrm{i}\mathrm{m}A=\left|Z(G_n)\bigcap G_n\right|{\left|G_n:Z(G_n)\right|}^{\mathrm{1}/\mathrm{2}}{\left|H:Z(H)\right|}^{\mathrm{1}/\mathrm{2}}/\left|H\right|$

Since information is stored exclusively on subsystem $A$, our concern is limited to errors that affect $A$. An error $E$ in $G_n$ is detectable by subsystem $A$ if and only if $E$ is contained in the set $E-(H{C}_{G_n}(H)-H)$. The distance of the subsystem code $Q$ is defined as

$d=\mathrm{m}\mathrm{i}\mathrm{n}\{\mathrm{w}\mathrm{t}(E):I\ne E\in H{C}_{G_n}(H)-H\}=\mathrm{w}\mathrm{t}(H{C}_{G_n}(H)-H).$

If $H{C}_{G_n}(H)=H$, then we define the distance of the subsystem code $Q$ to be $\mathrm{w}\mathrm{t}(H)$. A distance $d$ subsystem code $Q=A\otimes B$ with $\mathrm{d}\mathrm{i}\mathrm{m}A=K$, $\mathrm{d}\mathrm{i}\mathrm{m}B=R$ is often denoted as $({(n,K,R,d))}_q$ or $[{[n,k,r,d]]}_q$ if $K=q^k$ and $R=q^r$. We assert that $H$ is the gauge group of $Q$ and $Z(H)$ is its stabilizer. The gauge group acts trivially on $A$.

The following theorem, as presented in Ref.[1], demonstrates the relationship between subsystem codes and classical codes.

Theorem 1   Let $C$ be a classical additive subcode of $F_q^{\mathrm{2}n}$ such that $C\ne \{\mathrm{0}\}$ and let $D$ denote its subcode $D=C\bigcap C^{{\perp }_s}$. If $x=\left|C\right|$ and $y=\left|D\right|$, then there exists a subsystem code $Q=A\otimes B$ such that

(1) $\mathbf{d}\mathbf{i}\mathbf{m}\mathit{A}={\mathit{q}}^{\mathit{n}}/{(\mathit{x}\mathit{y})}^{\mathrm{1}/\mathrm{2}}$ ;

(2) $\mathrm{d}\mathrm{i}\mathrm{m}B={(xy)}^{\mathrm{1}/\mathrm{2}}$ .

The minimum distance of the subsystem $A$ is given by

(1) $\mathit{d}=\mathbf{s}\mathbf{w}\mathbf{t}(\mathit{C}+{\mathit{C}}^{{\perp }_{\mathit{s}}}-\mathit{C})=\mathbf{s}\mathbf{w}\mathbf{t}({\mathit{D}}^{{\perp }_{\mathit{s}}}-\mathit{C})$ if ${\mathit{D}}^{{\perp }_{\mathit{s}}}\ne \mathit{C}$;

(2) $d=\mathrm{s}\mathrm{w}\mathrm{t}(D^{{\perp }_s})$ if $D^{{\perp }_s}=C$. Thus, the subsystem $A$ can detect all errors in $E$ of weight less than $d$, and correct all errors in $E$ of weight $\le \lfloor (d-\mathrm{1})/\mathrm{2}\rfloor$ .

We call codes constructed using Theorem 1 as Clifford subsystem codes.

The subsequent lemma will play an essential role in constructing Clifford subsystem codes, as detailed in Ref. [2].

Theorem 2   Let $C_{\mathrm{1}}$ be an $[n,k_{\mathrm{1}}{]}_q$ linear code such that subcode $C_{\mathrm{2}}=C_{\mathrm{1}}\bigcap C_{\mathrm{1}}^{\perp }$ is an $[n,k_{\mathrm{2}}{]}_q$ linear code and $k_{\mathrm{1}}+k_{\mathrm{2}}<n$. Then there exist $[[n,n-(k_{\mathrm{1}}+k_{\mathrm{2}}),k_{\mathrm{1}}-k_{\mathrm{2}},\mathrm{w}\mathrm{t}(C_{\mathrm{2}}^{\perp }\backslash C_{\mathrm{1}}{)]]}_q$ Clifford subsystem codes.

Theorem 3   (Singleton Bound for Clifford Subsystem Codes^[10]) Let $C$ be a $[{[n,k,r,d]]}_q$ Clifford subsystem code. Then $\mathrm{2}d\le n-(k+r)+\mathrm{2}$.

Definition 1^[4] Let $C$ be a $[{[n,k,r,d]]}_q$ Clifford subsystem code. If $C$ attains Singleton bound for Clifford subsystem code, i.e., $\mathrm{2}d=n-(k+r)+\mathrm{2}$, it is termed a Clifford subsystem MDS code.

2 Constructions of Clifford Subsystem MDS Codes

In this section, we construct three classes of Clifford subsystem MDS codes employing cyclic codes and generalized Reed-Solomon codes over $F_q$.

Throughout the following, we consistently assume that $n$ is a positive integer.

A linear code of length n over $F_q$ is cyclic if the code invariant under the automorphism $\tau$ which

$\tau (c_{\mathrm{0}},c_{\mathrm{1}},\cdots ,c_{n-\mathrm{1}})=(c_{n-\mathrm{1}},c_{\mathrm{0}},\cdots ,c_{n-\mathrm{2}})$

It is well-known that a cyclic code of length $n$ over $F_q$ can be identified with an ideal in the residue ring $\frac{F_q[x]}{〈x^n-\mathrm{1}〉}$ via the isomorphism $\phi :F_q^n\to \frac{F_q[x]}{〈x^n-\mathrm{1}〉}$ given by $(a_{\mathrm{0}},a_{\mathrm{1}},\cdots ,a_{n-\mathrm{1}})\mapsto a_{\mathrm{0}}+a_{\mathrm{1}}{x}_{\mathrm{1}}+\cdots +a_{n-\mathrm{1}}{x}^{n-\mathrm{1}}(\mathrm{m}\mathrm{o}\mathrm{d}(x^n-\mathrm{1}))$. From that, the following fact is well-known and straightforward (see Ref.[8]).

Lemma 1   If $C$ is a cyclic code of length $n$ over $F_q$, then there exists $f(x)\in F_q[x]$ such that $C=〈f(x)〉$ with $f(x)|x^n-\mathrm{1}$.

Let $i$ be an integer such that $\mathrm{0}\le i\le n-\mathrm{1}$, and let $l$ be the smallest positive integer such that $i{q}^l\equiv i(\mathrm{m}\mathrm{o}\mathrm{d}n)$. Then $C_i=\{i,iq,\cdots ,i{q}^{l-\mathrm{1}}\}$ is the $q$-cyclotomic coset module $n$ containing $i$. Since $q$ is coprime with $n$, the fundamental factors of $x^n-\mathrm{1}$ in $F_q[x]$ can be described by the $q$-cyclotomic cosets. Suppose that $\alpha$ be a primitive nth root of unity over some extension field of $F_q$, and let $M_j(x)$ be the minimal polynomial of ${\alpha }^j$ concerning $F_q$. Let $\{s_{\mathrm{1}},s_{\mathrm{2}},\cdots ,s_t\}$ be a complete set of representatives of $q$-cyclotomic cosets. Then, the polynomial $x^n-\mathrm{1}$ factors uniquely into monic irreducible polynomial in $F_q[x]$ as $x^n-\mathrm{1}={\prod }_{j=\mathrm{1}}^t{M}_{s_j}(x)$.

The defining set of the cyclic code $C=〈f(x)〉$ is defined as $Z(C)=\{i\in Z_n|f({\alpha }^i)=\mathrm{0}\}$.

The defining set $Z(C)$ is a union of some $q$-cyclotomic co-sets and $\mathrm{d}\mathrm{i}\mathrm{m}(C)=n-\left|Z(C)\right|$.

Next, we recall some primary results of generalized Reed-Solomon codes (see Ref.[7]). Let ${\alpha }_{\mathrm{1}},\cdots ,{\alpha }_n$ be $n$ distinct elements of $F_q$, and let $v_{\mathrm{1}},\cdots ,v_n$ be $n$ nonzero elements of $F_q$. For $k$ between 1 and $n$, the generalized Reed-Solomon code $\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v})$ is defined by

$\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v})=\{(v_{\mathrm{1}}f({\alpha }_{\mathrm{1}}),\cdots ,v_{n}f({\alpha }_n))|f(x)\in F_q[x],\mathrm{d}\mathrm{e}\mathrm{g}(f(x))\le k-\mathrm{1}\}$

where $\mathit{a},\mathit{v}$ denote the vectors $({\alpha }_{\mathrm{1}},\cdots ,{\alpha }_n),(v_{\mathrm{1}},\cdots ,v_n)$, respectively.

2.1 Construction 1

Let $a\ge \mathrm{0}$ and $\mathrm{1}\le k\le q-\mathrm{1}$. A Reed-Solomon code (RS code) is a cyclic code of length $q-\mathrm{1}$ generated by

$g(x)=(x-{\alpha }^a)(x-{\alpha }^{a+\mathrm{1}})\cdots (x-{\alpha }^{a+n-k-\mathrm{1}})$

denoted by $\mathrm{R}{\mathrm{S}}_k(n,a)$, where $\alpha$ is a primitive element of $F_q$ (see Ref.[11]).

Remark 1   It is easy to prove that $\mathrm{R}{\mathrm{S}}_k{(n,a)}^{\perp }=\mathrm{R}{\mathrm{S}}_{n-k}(n,n-a+\mathrm{1})$. Thus, the defining set of $\mathrm{R}{\mathrm{S}}_k{(n,a)}^{\perp }$ is given by $Z(\mathrm{R}{\mathrm{S}}_k{(n,a)}^{\perp })=\{n-a+\mathrm{1},n-a+\mathrm{2},\cdots ,n-a+k\}$.

Lemma 2   Let $C$ be cyclic code with defining set $Z(C)$. Then the defining set of $C\bigcap C^{\perp }$ is given by $Z(C)\bigcup Z(C^{\perp })$. In particular, $\mathrm{d}\mathrm{i}{\mathrm{m}}_{F_q}(C\bigcap C^{\perp })=\left|Z(C)\bigcap Z(C^{\perp })\right|$.

Proof   According to Ref.[6] (Exercise 239, Chapter 4), we have that $Z(C\bigcap C^{\perp })=Z(C)\bigcup Z(C^{\perp })$. Thus,

$\mathrm{d}\mathrm{i}{\mathrm{m}}_{F_q}(C\bigcap C^{\perp })=n-\left|Z(C)\bigcup Z(C^{\perp })\right|=n-\left|Z(C)\right|-\left|Z(C^{\perp })\right|+\left|Z(C)\bigcap Z(C^{\perp })\right|=\left|Z(C)\bigcap Z(C^{\perp })\right|$

Theorem 4   Let $n=q-\mathrm{1},a\ge \mathrm{1}$, and $\mathrm{2}a-\mathrm{1}\le k<\lceil \frac{n+\mathrm{2}a-\mathrm{1}}{\mathrm{2}}\rceil$. Then, there is a Clifford subsystem MDS code with parameters $[{[n,n-\mathrm{2}k+\mathrm{2}a-\mathrm{1,2}a-\mathrm{1},k-\mathrm{2}a+\mathrm{2}]]}_q$.

Proof   Let $C=\mathrm{R}{\mathrm{S}}_k(n,a)$. Obviously, the defining set of $C$ is given by $Z(C)=\{a,a+\mathrm{1},\cdots ,n+a-k-\mathrm{1}\}$. By Remark 1, we have $Z(C^{\perp })=\{n-a+\mathrm{1},n-a+\mathrm{2},\cdots ,n-a+k\}$}.

Since $k\ge \mathrm{2}a-\mathrm{1}$, we have $n+a-k-\mathrm{1}\le n-a<n-a+\mathrm{1}$. Then, the first element in the defining set of $Z(C^{\perp })$ comes after the last element in $Z(C)$. Since $a\ge \mathrm{1},k\ge \mathrm{2}a-\mathrm{1}\ge a$, we rewrite $Z(C^{\perp })$ as $Z(C^{\perp })=\{-a+\mathrm{1},-a+\mathrm{2},\cdots ,$$-\mathrm{1,0},\mathrm{1},\cdots ,k-a\}$. Then $\left|Z(C)\bigcap Z(C^{\perp })\right|=k-\mathrm{2}a+\mathrm{1}$. And $Z(C)\bigcup Z(C^{\perp })=\{-a+\mathrm{1},-a+\mathrm{2},\cdots ,-\mathrm{1,0},\mathrm{1},\cdots ,k-a,\cdots ,n+a-k-\mathrm{1}\}$.

This means that $D=C\bigcap C^{\perp }$ is an MDS code with parameters ${[n,k-\mathrm{2}a+\mathrm{1},n-k+\mathrm{2}a]}_q$. It follows that $D^{\perp }$ is an MDS code with parameters ${[n,n-k+\mathrm{2}a-\mathrm{1},k-\mathrm{2}a+\mathrm{2}]}_q$.

By $k<\lceil \frac{n+\mathrm{2}a-\mathrm{1}}{\mathrm{2}}\rceil$, we have $d(D^{\perp }\backslash C)=k-\mathrm{2}a+\mathrm{2}$. Then, by Theorem 2, there exists a Clifford subsystem code $Q$ with parameters $[{[n,n-\mathrm{2}k+\mathrm{2}a-\mathrm{1,2}a-\mathrm{1},k-\mathrm{2}a+\mathrm{2}]]}_q$.

Since $\mathrm{2}(k-\mathrm{2}a+\mathrm{2})=n-((n-\mathrm{2}k+\mathrm{2}a-\mathrm{1})+(\mathrm{2}a-\mathrm{1}))+\mathrm{2}$, the Clifford subsystem code $Q$ with parameters $[{[n,n-\mathrm{2}k+\mathrm{2}a-\mathrm{1,2}a-\mathrm{1},k-\mathrm{2}a+\mathrm{2}]]}_q$ is MDS by Definition 1.

2.2 Construction 2

In this subsection, we construct a class of Clifford subsystem MDS codes by using generalized Reed-Solomon codes over $F_q$.

Let $n|q-\mathrm{1}$, and let $\alpha$ be a nth root of unity, that is ${\alpha }^n=\mathrm{1}$ and ${\alpha }^i\ne \mathrm{1}$ for $\mathrm{1}\le i<n$. Take ${\mathit{a}}_{\mathrm{1}}=(\mathrm{1},\alpha ,{\alpha }^{\mathrm{2}},\cdots ,{\alpha }^{n-\mathrm{1}})$ and ${\mathit{v}}_{\mathrm{1}}=\mathrm{1}$ with $\mathrm{1}=(\mathrm{1,1},\cdots ,\mathrm{1})$; the generalized Reed-Solomon code $\mathrm{G}\mathrm{R}{\mathrm{S}}_k({\mathit{a}}_{\mathrm{1}},{\mathit{v}}_{\mathrm{1}})$ have the following generator matrix:

${\mathit{G}}_{\mathrm{G}\mathrm{R}{\mathrm{S}}_k({\mathit{a}}_{\mathrm{1}},{\mathit{v}}_{\mathrm{1}})}=\left(\begin{array}{ccccc}\mathrm{1}& \mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}\\ \mathrm{1}& \alpha & {\alpha }^{\mathrm{2}}& \cdots & {\alpha }^{n-\mathrm{1}}\\ \mathrm{1}& {\alpha }^{\mathrm{2}}& {\alpha }^{\mathrm{2}(\mathrm{2})}& \cdots & {\alpha }^{\mathrm{2}(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{1}& {\alpha }^{k-\mathrm{1}}& {\alpha }^{(k-\mathrm{1})\mathrm{2}}& \cdots & {\alpha }^{(k-\mathrm{1})(n-\mathrm{1})}\end{array}\right)$

where $\mathrm{1}\le k\le n$.

The rows of the matrix ${\mathit{G}}_{\mathrm{G}\mathrm{R}{\mathrm{S}}_n({\mathit{a}}_{\mathrm{1}},{\mathit{v}}_{\mathrm{1}})}$ under consideration will be denoted by $\{{\mathit{g}}_{\mathrm{0}},{\mathit{g}}_{\mathrm{1}},\cdots ,{\mathit{g}}_{n-\mathrm{1}}\}$.

Thus ${\mathit{g}}_j=(\mathrm{1},{\alpha }^j,{\alpha }^{j(\mathrm{2})},\cdots ,{\alpha }^{j(n-\mathrm{1})})$ for $j=\mathrm{0,1},\cdots ,n-\mathrm{1}$. It is easy to check that ${\mathit{g}}_i{\mathit{g}}_j^{\mathrm{T}}=\mathrm{0}$ for $j\ne n-i$. We recall the following fact (see Ref.[12]).

Lemma 3   Let $C$ be a code generated by taking $k$ consecutive rows of the matrix ${\mathit{G}}_{\mathrm{G}\mathrm{R}{\mathrm{S}}_n({\mathit{a}}_{\mathrm{1}},{\mathit{v}}_{\mathrm{1}})}$. Then $C$ is an MDS code with parameters ${[n,k,n-k+\mathrm{1}]}_q$.

It is a routine to verify the following lemma.

Lemma 4   Let $C$ be the code with generator matrix $\mathit{G}=\left(\begin{array}{c}{\mathit{g}}_{\mathrm{0}}\\ {\mathit{g}}_{\mathrm{1}}\\ ⋮\\ {\mathit{g}}_{k-\mathrm{1}}\end{array}\right)$. Then $\mathit{H}=\left(\begin{array}{c}{\mathit{g}}_{\mathrm{1}}\\ {\mathit{g}}_{\mathrm{2}}\\ ⋮\\ {\mathit{g}}_{n-k}\end{array}\right)$ is a check matrix for $C$.

Theorem 5   Let $n|q-\mathrm{1},\mathrm{0}<k<\lceil \frac{n+\mathrm{1}}{\mathrm{2}}\rceil$. Then, there is a Clifford subsystem MDS code $Q$ with parameters $[{[n,n-\mathrm{2}k+\mathrm{1,1},k]]}_q$.

Proof   For $\mathrm{0}<k<\lceil \frac{n+\mathrm{1}}{\mathrm{2}}\rceil$, let

${\mathit{G}}_{\mathrm{1}}=\left(\begin{array}{ccccc}\mathrm{1}& \mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}\\ \mathrm{1}& \alpha & {\alpha }^{\mathrm{2}}& \cdots & {\alpha }^{n-\mathrm{1}}\\ \mathrm{1}& {\alpha }^{\mathrm{2}}& {\alpha }^{\mathrm{2}(\mathrm{2})}& \cdots & {\alpha }^{\mathrm{2}(n-\mathrm{1})}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ \mathrm{1}& {\alpha }^{k-\mathrm{1}}& {\alpha }^{(k-\mathrm{1})\mathrm{2}}& \cdots & {\alpha }^{(k-\mathrm{1})(n-\mathrm{1})}\end{array}\right)=\left(\begin{array}{c}{\mathit{g}}_{\mathrm{0}}\\ {\mathit{g}}_{\mathrm{1}}\\ ⋮\\ {\mathit{g}}_{k-\mathrm{1}}\end{array}\right)$

The code $C_{\mathrm{1}}$ generated by the matrix ${\mathit{G}}_{\mathrm{1}}$ is an MDS code with parameters ${[n,k,n-k+\mathrm{1}]}_q$ by Lemma 3. Taking ${\mathit{H}}_{\mathrm{1}}=\left(\begin{array}{c}{\mathit{g}}_{\mathrm{1}}\\ {\mathit{g}}_{\mathrm{2}}\\ ⋮\\ {\mathit{g}}_{n-k}\end{array}\right)$. Then, by Lemma 4, ${\mathit{H}}_{\mathrm{1}}$ is a generator of the matrix of code $C_{\mathrm{1}}^{\perp }$ with parameters ${[n,n-k,k+\mathrm{1}]}_q$.

Let ${\mathit{C}}_{\mathrm{2}}=C_{\mathrm{1}}\bigcap C_{\mathrm{1}}^{\perp }$. Set ${\mathit{G}}_{\mathrm{2}}=\left(\begin{array}{c}{\mathit{g}}_{\mathrm{1}}\\ {\mathit{g}}_{\mathrm{2}}\\ ⋮\\ {\mathit{g}}_{k-\mathrm{1}}\end{array}\right)$. Then, by $k<\lceil \frac{n+\mathrm{1}}{\mathrm{2}}\rceil$, the matrix ${\mathit{C}}_{\mathrm{2}}$ is a generator matrix for code $C_{\mathrm{2}}$. Moreover, the code $C_{\mathrm{2}}$ is an MDS code with parameters ${[n,k-\mathrm{1},n-k+\mathrm{2}]}_q$ by Lemma 3. So, the code $C_{\mathrm{2}}^{\perp }$ is an MDS code with parameters ${[n,n-k+\mathrm{1},k]}_q$.

By Theorem 2, there exists a Clifford subsystem code $Q$ with parameters $[{[n,n-\mathrm{2}k+\mathrm{1,1},k]]}_q$.

Since $\mathrm{2}d=\mathrm{2}k=n-(n-\mathrm{2}k+\mathrm{2})+\mathrm{2}$, the Clifford subsystem code $Q$ with parameters $[{[n,n-\mathrm{2}k+\mathrm{1,1},k]]}_q$ is MDS by Definition 1.

2.3 Construction 3

In this subsection, we construct a class of Clifford subsystem MDS codes by utilizing the extended code of generalized Reed-Solomon codes over $F_q$.

We note that the extended code of the generalized Reed-Solomon code $\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v})$ given by

$\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v},\mathrm{\infty })=\{v_{\mathrm{1}}f({\alpha }_{\mathrm{1}}),v_{\mathrm{2}}f({\alpha }_{\mathrm{2}}),\cdots ,v_{n}f({\alpha }_n),f_{k-\mathrm{1}}\}:f(x)\in F_q[x],\mathrm{d}\mathrm{e}\mathrm{g}(f(x))\le k-\mathrm{1}\}$

where $f_{k-\mathrm{1}}$ stands for the coefficient of $x^{{}_{k-\mathrm{1}}}$ . The following two results can be found in Ref.[7].

Lemma 5^[13] The code $\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v},\mathrm{\infty })$ is an MDS code with parameters ${[n+\mathrm{1},k,n-k+\mathrm{2}]}_q$.

Lemma 6 ^[13] Let 1 be all-one word of length $n$. If $\mathrm{1}\le k\le q-\mathrm{1}$, then the dual code of $\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathrm{1},\mathrm{\infty })$ is $\mathrm{G}\mathrm{R}{\mathrm{S}}_{q-k+\mathrm{1}}(\mathit{a},\mathrm{1},\mathrm{\infty })$.

Theorem 6   Let $\mathrm{1}\le k<\lceil \frac{q+\mathrm{1}}{\mathrm{2}}\rceil$ . Then, there is a Clifford subsystem MDS code with parameters $[{[q+\mathrm{1},q-\mathrm{2}k+\mathrm{2,1},k]]}_q$.

Proof   Taking $\mathit{G}=\left(\begin{array}{ccccc}\mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}& \mathrm{0}\\ {\alpha }_{\mathrm{1}}& {\alpha }_{\mathrm{2}}& \cdots & {\alpha }_q& \mathrm{0}\\ {\alpha }_{\mathrm{1}}^{\mathrm{2}}& {\alpha }_{\mathrm{2}}^{\mathrm{2}}& \cdots & {\alpha }_q^{\mathrm{2}}& \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\alpha }_{\mathrm{1}}^{k-\mathrm{1}}& {\alpha }_{\mathrm{2}}^{k-\mathrm{1}}& \cdots & {\alpha }_q^{k-\mathrm{1}}& \mathrm{1}\end{array}\right)$. Then, $\mathit{G}$ is a generator matrix of the code $C=\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v},\mathrm{\infty })$ with parameters ${[q+\mathrm{1},k,q-k+\mathrm{2}]}_q$.

Set $\mathit{H}=\left(\begin{array}{ccccc}\mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}& \mathrm{0}\\ {\alpha }_{\mathrm{1}}& {\alpha }_{\mathrm{2}}& \cdots & {\alpha }_q& \mathrm{0}\\ {\alpha }_{\mathrm{1}}^{\mathrm{2}}& {\alpha }_{\mathrm{2}}^{\mathrm{2}}& \cdots & {\alpha }_q^{\mathrm{2}}& \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\alpha }_{\mathrm{1}}^{q-k}& {\alpha }_{\mathrm{2}}^{q-k}& \cdots & {\alpha }_q^{q-k}& \mathrm{1}\end{array}\right)$ . Then, by Lemma 6, $\mathit{H}$ is a parity-check matrix of the code $C=\mathrm{G}\mathrm{R}{\mathrm{S}}_k(\mathit{a},\mathit{v},\mathrm{\infty })$.

Since $\mathrm{1}\le k<\lceil \frac{q+\mathrm{1}}{\mathrm{2}}\rceil$, ${\mathit{G}}_{C\bigcap C^{\perp }}=\left(\begin{array}{ccccc}\mathrm{1}& \mathrm{1}& \cdots & \mathrm{1}& \mathrm{0}\\ {\alpha }_{\mathrm{1}}& {\alpha }_{\mathrm{2}}& \cdots & {\alpha }_q& \mathrm{0}\\ {\alpha }_{\mathrm{1}}^{\mathrm{2}}& {\alpha }_{\mathrm{2}}^{\mathrm{2}}& \cdots & {\alpha }_q^{\mathrm{2}}& \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {\alpha }_{\mathrm{1}}^{k-\mathrm{2}}& {\alpha }_{\mathrm{2}}^{k-\mathrm{2}}& \cdots & {\alpha }_q^{k-\mathrm{2}}& \mathrm{0}\end{array}\right)$ is a generator matrix for the code $D=C\bigcap C^{\perp }$ with parameters ${[q+\mathrm{1},k-\mathrm{1},q-k+\mathrm{1}]}_q$. It is easy to check that $D^{\perp }$ is a linear code with parameters ${[q+\mathrm{1},q-k+\mathrm{2},k]}_q$.

Since $\mathrm{1}\le k<\lceil \frac{q+\mathrm{1}}{\mathrm{2}}\rceil$, $q-k+\mathrm{2}>k$, which implies that $d(D^{\perp }\backslash C)=k$. Thus, by Theorem 2, there exists a Clifford subsystem code $Q$ with parameters $[{[q+\mathrm{1},q-\mathrm{2}k+\mathrm{2,1},k]]}_q$.

Since $\mathrm{2}k=q+\mathrm{1}-(q-\mathrm{2}k+\mathrm{2}+\mathrm{1})+\mathrm{2}$, the Clifford subsystem code $Q$ with parameters $[{[q+\mathrm{1},q-\mathrm{2}k+\mathrm{2,1},k]]}_q$ is MDS by Definition 1.

3 Comparison and Conclusion

This paper presents three new families of Clifford subsystem MDS codes employing Reed-Solomon codes and extended Reed-Solomon codes over $F_q$. Table 1 gives our general conclusions to compare those known results in Refs. [5-7]. The results show that the lengths of those known conclusions above Clifford subsystem codes studied in the pieces of literature are fixed or odd. However, the lengths of a class of Clifford subsystem codes derived from our construction are very flexible.

References

All Tables