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

Abstract

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 $[[n,n-2d+2,d]]_q$ codes with minimum distances $d>\frac{q}{2}$ for sparse lengths $n>q+1$. In the case $n=\frac{q^2-1}{m}$ where $m|q+1$ or $m|q-1$ there are complete results. In the case $n=\frac{q^2-1}{m}$ while $m|q^2-1$ is neither a factor of $q-1$ nor $q+1$, no q-ary quantum MDS code with $d> \frac{q}{2}$ has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over $\mathbf{F}_{q^2}$. 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 $\frac{w(q^2-1)}{u}$ and minimum distances $d > \frac{q}{2}$ are presented.

1 Introduction

Quantum error-correcting codes are important for quantum information processing and quantum computation. The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of [15, 19, 20]. It is known for any pure quantum $[[n, k, d]]_q$ code the parameters satisfy the quantum singleton bound $k \le n-2d+2$. The q-ary quantum codes reaching this bound are called quantum MDS codes [2, 14, 15]. Many constructions of q-ary quantum MDS codes have been proposed based on the Hermitian self-orthogonal codes over $\mathbf{F}_{q^2}$.

The Hermitian inner product over $\mathbf{F}_{q^2}^n$ is defined as follows. $<\mathbf{u},\mathbf{v}>_h=u_1v_1^q+\cdots +u_nv_n^q$, where $\mathbf{u}=(u_1,\ldots ,u_n)$ and $\mathbf{v}=(v_1,\ldots ,v_n)$ are vectors in $\mathbf{F}_{q^2}^n$. The following result gives a construction of q-ary quantum MDS codes from Hermitian self-orthogonal MDS codes over $\mathbf{F}_{q^2}$.

Theorem 1.1

([2]) If $\mathbf{C}$ is a $[n,k,n-k+1]_{q^2}$ MDS code over $\mathbf{F}_{q^2}$ which is orthogonal under the Hermitian inner product. Then we have a q-ary quantum MDS $[[n, n-2k, k+1]]_q$ code.

There have been published many papers on the construction of quantum MDS codes [1, 2, 4–17]. They were constructed from generalized Reed–Solomon codes [8–10], cyclic or constacyclic codes [3, 7, 11, 12]. However, it seems that for many lengths $q+1<n< q^2-1$ whether there is a q-ary quantum MDS code with length n and minimum distance $d>\frac{q}{2}$ is still an un-solved problem. For only very few sparse lengths such q-ary quantum MDS codes with $d>\frac{q}{2}$ have been constructed [3, 7–12, 21]. In the case of length $n=\frac{q^2-1}{m}$ where m is an integer satisfying $m|q+1$ or $m|q-1$ the following results have been proved ([3, 13, 21], or see lines 13, 14 and 20 in the table of page 1482 of [3]).

• 1. For odd prime powers $q=2^es+1$ where s is odd, an odd factor $\lambda |s$ of s and $f \le e-1$, a quantum MDS $[[2^f\lambda (q+1), 2^f\lambda (q+1)-2d+2, d]]_q$ code with minimum distance d for each integer d in the range $2 \le d \le \frac{q+1}{2}+2^f\lambda$ was constructed ([3] Theorem 4.11).

• 2. In the case $m|q+1$ and m odd there is a q-ary quantum MDS code with length $\frac{q^2-1}{m}$ and minimum distance d for each integer d in the range $2 \le d \le \frac{q+1}{2}+\frac{q+1}{2m}-1$. In the case $m|q+1$ and m even there is a q-ary quantum MDS code with length $\frac{q^2-1}{m}$ and minimum distance d for each integer d in the range $2 \le d \le \frac{q+1}{2}+\frac{q+1}{m}-1$ ( see [3, 21]).

However, in the case $n=\frac{q^2-1}{m}$ where $m|q^2-1$ is neither a factor of $q-1$ nor $q+1$, no q-ary quantum MDS code with length $\frac{q^2-1}{m}$ and minimum distance $d > \frac{q}{2}$ has been constructed. Though in this case each cyclotomic set has only one element, the technique in [3, 8, 12, 13] is not sufficient to get the desirable q-ary quantum MDS codes. In this paper some new q-ary quantum MDS codes in this case with minimum distance $d>\frac{q}{2}$ are constructed. We use a direct approach of constructing Hermitian self-orthogonal MDS codes over $\mathbf{F}_{q^2}$. Many new q-ary quantum MDS codes for the length $n=\frac{w(q^2-1)}{u}$ and $d>\frac{q}{2}$ for some integers w and u are also presented.

We need the following lemmas in this paper.

Lemma 1.1

If $\theta$ is a primitive element of the multiplicative group $\mathbf{F}_{q^2}^{*}$ and suppose m is a factor of $q^2-1$, then $\Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{jtm}=0$ except the case that t is divisible by $\frac{q^2-1}{m}$.

Proof

For any $1 \le t \le \frac{q^2-1}{m}-1$, $\theta ^{mt}$ generates a subgroup G of the group $\mathbf{Z}/(\frac{q^2-1}{m})\mathbf{Z}$ generated by $\theta ^{m}$. The order of the group G is $\frac{\frac{q^2-1}{m}}{gcd(t, \frac{q^2-1}{m})} >1$. Since $G \ne \{1\}$, for any non-unit element $\theta ^{mt}$, $\theta ^{mt} G=G$. Thus $\theta ^{mt} \Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{mtj} = \Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{mtj}$. It is clear $\theta ^{mt} \ne 1$ when t is not divisible by $\frac{q^2-1}{m}$. The conclusion follows directly. $\square$

Lemma 1.2

Suppose $v_1, \ldots ,v_n$ are n nonzero elements in the multiplicative group $\mathbf{F}_q^*$. If $\mathbf{g}_l=(g_{1l},\ldots ,g_{nl})$ where $l=1,\ldots ,k$, are k linear independent rows in $\mathbf{F}_{q^2}^n$ satisfying that $\Sigma _{j=1}^n v_j g_{jl_1} g_{jl_2}^q=0$ for any two indices $l_1$ and $l_2$ in the set $\{1,\ldots ,k\}$ (here $l_1=l_2$ is possible). Then we have a Hermitian self-orthogonal $[n, k]_{q^2}$ code generated by these k rows.

Proof

We can set $v_j=(v_j^{\prime })^{q+1}$ for $j=1,\ldots ,n$. Thus the equivalent code $(v_1^{\prime },\ldots ,v_n^{\prime }) \mathbf{C}$ is a Hermitian self-orthogonal code, where $\mathbf{C}$ is a $q^2$-ary code generated by these k rows $\mathbf{g}_1,\ldots ,\mathbf{g}_k$.

The main idea to construct Hermitian self-orthogonal codes in this paper is as follows. It is well known that from Lemma 1.1 we can prove that the dual of a Reed–Solomon code (evaluation vectors of all polynomials with degrees less than k at a subset $\mathbf{S}$ of $\mathbf{F}_{q^2}$) is another Reed–Solomon code (evaluation vectors of all polynomials with degrees less than $|\mathbf{S}|-k$ at this subset $\mathbf{S}$ of $\mathbf{F}_{q^2}$) (see [18]). Hence we only need to guarantee the condition of Lemma 1.1 is satisfied so that Hermitian self-orthogonal MDS codes can be constructed. There are q-th powers in the Hermitian inner product $\Sigma _{i=1}^n u_i v_i^q$. For the purpose to enlarge dimensions of constructed Hermitian self-orthogonal MDS codes, we need some number theoretical conditions on the lengths to guarantee that the exponential sums in the Hermitian inner products are zero. Then q-ary quantum MDS codes with minimum distances bigger than $\frac{q}{2}$ can be constructed. $\square$

2 New quantum MDS codes I

2.1 Construction 1

Let m be a factor of $q^2-1$. For any fixed positive integer w we define a length $\frac{q^2-1}{m}$ linear error code over $\mathbf{F}_{q^2}$ as follows.

$$\begin{aligned} \mathbf{C}_w= & {} \{(\theta ^{m} f(\theta ^{m}), \theta ^{2m}f(\theta ^{2m}),\ldots ,\theta ^{jm}f(\theta ^{jm}),\ldots ,\\&\theta ^{(\frac{q^2-1}{m}-1)m} f(\theta ^{(\frac{q^2}{m}-1)m}), f(1)): f \in \mathbf{F}_{q^2}[x],\deg (f) \le w-1\}\\ \end{aligned}$$

It is clear that $\mathbf{C}_w$ is a MDS $[\frac{q^2-1}{m}, w, \frac{q^2-1}{m}-w+1]$ code over $\mathbf{F}_{q^2}$. Actually this code is equivalent to a evaluation code at the elements $\theta ^m, \theta ^{2m},\ldots$, $\theta ^{(\frac{q^2-1}{m}-1)m}, 1$. Hence it is equivalent to a Reed–Solomon code.

The Hermitian inner product of any two codewords (corresponding to two polynomials f and g) is $\Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{jm+jqm} fg^q(\theta ^{jm})$. Thus we only need to check

$$\begin{aligned} \Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{(q+1)mj} \theta ^{jm(t_1+t_2q)}=\Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{jm(q+1+t_1+t_2q)}=0, \end{aligned}$$

where $0 \le t_1, t_2 \le w-1$.

Theorem 2.1

If $m=2k+1$ is an odd positive factor of $q+1$ and $w < \frac{k+1}{2k+1}(q-1)$, then for all non-negative integers $t_1$ and $t_2$ satisfying $0 \le t_1, t_2 \le w-1$, $q+1+t_1+t_2q$ is not divisible by $\frac{q^2-1}{m}$. Hence the code $\mathbf{C}_w$ is Hermitian self-orthogonal.

Proof

It is clear that if $\Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{jm(q+1+t_1+t_2q)}=0$ for all $t_1$ and $t_2$ satisfying $0 \le t_1, t_2 \le w-1$, the code is Hermitian self-orthogonal. Hence from Lemma 1.1 it is sufficient to prove that if $w <\frac{k+1}{2k+1}(q-1)$, $q+1+t_1+t_2q$, where $t_1<w,t_2<w$, is not divisible by $\frac{q^2-1}{m}$. Since $q+1+t_1+t_2q \le (q+1)(1+w-1) <(k+1)\frac{q^2-1}{m}$, if $q+1+t_1+t_2q$ is divisible by $\frac{q^2-1}{m}$, the quotient $\frac{q+1+t_1+t_2q}{\frac{q^2-1}{m}} \le k$. On the other hand $\frac{q^2-1}{m}=\frac{q+1}{m}q-\frac{q+1}{m}$. That is, $\frac{q^2-1}{m} \equiv q-\frac{q+1}{m}$ mod q because $\frac{q+1}{m}$ is an integer. Therefore, if $q+1+t_1+t_2q$ is divisible by $\frac{q^2-1}{m}$, then residue of $q+1+t_1+t_2q$ module q is in the range $[\frac{k+1}{m}(q+1)-1, q-1]$. It is obvious that the residue of $q+t_2q+1+t_1$ module q is $1+t_1 \le w <\frac{k+1}{2k+1}(q-1)$. Since $\frac{k+1}{m}(q+1)$ is a positive integer and $\frac{k+1}{m}(q-1)=\frac{k+1}{m}(q+1)-1-\frac{1}{m}<\frac{k+1}{m}(q+1)$, the conclusion follows directly. $\square$

Corollary 2.1

If $m=2k+1$ is an odd factor of $q+1$, for each positive integer d in the range $2\le d \le \lfloor \frac{k+1}{2k+1}(q-1)+1 \rfloor$, there exists a q-ary quantum MDS code with length $\frac{q^2-1}{m}$ and minimum distance d.

Suppose q is a prime power and $q+1=\lambda r$ where r is an odd integer, then for each integer d in the range $2 \le d \le \frac{q-1}{2}+\frac{\lambda }{2}$, a length $\lambda (q-1)$ q-ary quantum MDS code with the minimum distance d was constructed in [3, 12, 13, 21]. Its construction was based on constacyclic codes over $\mathbf{F}_{q^2}$. However, this kind of quantum q-ary MDS codes is a direct consequence from the constructed quantum MDS codes in Corollary 2.1.

We can extend the construction 1 to $[\frac{q^2-1}{m}+1, w+1, \frac{q^2-1}{m}-w+1]$ Hermitian self-orthogonal code over $\mathbf{F}_{q^2}$ with the following generator matrix.

$$\begin{aligned} \left( \begin{array}{ccccccccccccccc} \frac{q+1}{m}&{}1&{}\cdots &{}1&{}1\\ 0&{}\theta ^{m}&{}\cdots &{}\theta ^{\left( \frac{q^2-1}{m}-2\right) m}&{}\theta ^{\frac{q^2-1}{m}m}=1\\ \cdots &{}\cdots &{}\cdots &{}\cdots &{}\cdots \\ 0&{}\theta ^{im}&{}\cdots &{}\theta ^{\left( \frac{q^2-1}{m}-2\right) im}&{}1\\ \cdots &{}\cdots &{}\cdots &{}\cdots &{}\cdots \\ 0&{}\theta ^{wm}&{}\cdots &{}\theta ^{\left( \frac{q^2-1}{m}-2\right) wm}&{}1\\ \end{array} \right) \end{aligned}$$

Therefore, the following result which can be thought as a generalization of Theorem 4.4 of [11] is proved.

Theorem 2.2

For each odd number $m=2k+1$ satisfying $m|q+1$, we have a $[[\frac{q^2+m-1}{m}, \frac{q^2+m-1}{m}-2d,d+1]]_q$ quantum MDS code for each integer d in the range $2 \le d \le \lfloor \frac{k+1}{2k+1}(q-1)+1\rfloor$.

In the case $q+1$ is divisible by 3 we have a length $\frac{q^2-1}{3}+1=\frac{q^2+2}{3}$ q-ary quantum MDS code with minimum distance d for each integer d in the range $2 \le d \le \frac{2(q+1)}{3}$. This recovers the 2nd conclusion of Theorem 4.4 of [10]. Moreover if $5|q+1$, then we have a length $\frac{q^2-1}{5}+1=\frac{q^2+4}{5}$ q-ary quantum MDS code with the minimum distance d for each integer d in the range $2 \le d \le \frac{3(q+1)}{5}$. We list some new quantum MDS codes from Theorem 2.2 in Table 1.

Table 1 $[[\frac{q^2+m-1}{m}, \frac{q^2+m-1}{m}-2d, d+1]]_q$ quantum MDS codes

2.2 Construction 2

We need the following two lemmas in construction 2. The main idea of the construction 2 is the consideration of the sum of two identities as in Lemma 1.1 with respect to two subsets. Then we have new identities that some exponential sums at a new subset are zero. This leads to some new Hermitian self-orthogonal codes with different lengths.

Lemma 2.1

Suppose q is an even prime power $2^h$. Let $\theta \in \mathbf{F}_{q^2}$ be a primitive element which generate the multiplicative group $\mathbf{F}_{q^2}^{*}$. If $m_1$ and $m_2$ are factors of $q^2-1$ satisfying $\gcd (m_1,m_2)=1$. We set $m_3=\frac{q^2-1}{m_1}$ and $m_4=\frac{q^2-1}{m_2}$. Let $\mathbf{M}_1$ be the set of all indices j satisfying $1 \le j \le m_3-1$ and j is not divisible by $m_2$, and $\mathbf{M}_2$ be the set of all indices j satisfying $1 \le j \le m_4-1$ and j is not divisible by $m_1$. Then $\Sigma _{j \in \mathbf{M}_1} \theta ^{m_1tj}+\Sigma _{j \in \mathbf{M}_2} \theta ^{m_2tj}=0$ for $t=1,\ldots ,min\{m_3,m_4\}-1$.

Proof

From Lemma 1.1 and the fact $-1=1$ in the finite field $\mathbf{F}_{2^{2h}}$ we get the conclusion. Here we should note that in the two equalities $\Sigma _{j=1}^{m_3} \theta ^{m_1tj}=0$ and $\Sigma _{j=1}^{m_4} \theta ^{m_2tj}=0$. The common part is $\Sigma _{j=1}^{m_5} \theta ^{m_1m_2tj}$, where $m_5=\frac{q^2-1}{m_1m_2}$. $\square$

Here $|\mathbf{M}_1|=m_3-\frac{q^2-1}{m_1m_2}$ and $|\mathbf{M}_2|=m_4-\frac{q^2-1}{m_1m_2}$.

Similarly we have the following Lemma 2.2. Suppose q is an even prime power $2^h$. Let $\theta \in \mathbf{F}_{q^2}$ be a primitive element which generate the multiplicative group $\mathbf{F}_{q^2}^{*}$. Consider $m_1, \ldots ,m_s$ factors of $q^2-1$ satisfying $\gcd (m_{s_1},m_{s_2})=1$ for any $s_1 \ne s_2$. We set $m_1^{\prime }=\frac{q^2-1}{m_1},\ldots , m_s^{\prime }=\frac{q^2-1}{m_s}$. Set $\mathbf{M}_u$ the subgroup of the multiplicative group $\mathbf{F}_{q^2}^*$ generated by $\theta ^{m_u}$. Let $\mathbf{M}_{s_1,\ldots ,s_l}$ be the intersection of $\mathbf{M}_{s_1},\ldots ,\mathbf{M}_{s_l}$ for distinct indices $s_1, \ldots , s_l$ in the set $\{1,\ldots ,s\}$. The set $\mathbf{M}$ is defined as the set of elements in $\mathbf{M}_1 \cup \cdots \cup \mathbf{M}_s$ by deleting these elements in the set $\mathbf{M}_{s_1, \ldots ,s_l}$ where l is even. The elements in $\mathbf{M}_{s_1,\ldots ,s_{l'}}$ where $l'$ is odd are remained.

Lemma 2.2

$\Sigma _{j \in \mathbf{M} \cap \mathbf{M}_1} \theta ^{m_1tj}+\Sigma _{j \in \mathbf{M}_2 \cap \mathbf{M}} \theta ^{jm_2t}+\cdots +\Sigma _{j \in \mathbf{M} \cap \mathbf{M}_s} \theta ^{jm_2t}=0$ for $t=1, \ldots ,min\{m_1^{\prime },\ldots ,m_s^{\prime } \}-1$.

If q be an even prime power $2^h$, $m_1=2k_1+1 <m_2=2k_2+1$ are odd factors of $q+1$ satisfying $\gcd (m_1,m_2)=1$. Set $m_3=\frac{q^2-1}{m_1}$, $m_4=\frac{q^2-1}{m_2}$, $M=m_3+m_4-\frac{2(q^2-1)}{m_1m_2}$. We construct a length M linear code $\mathbf{C_M}$ over $\mathbf{F}_{q^2}$ as follows. $\mathbf{C_M}=\{(xf(x))_{x \in \mathbf{M}}: 0 \le \deg (f) \le w-1\}$, where $w <\frac{k_2+1}{2k_2+1} (q-1)$. This is equivalent to a evaluation code (a Reed–Solomon code) at all elements of the set $\mathbf{M}$. Thus this is a $[M, w, M-w+1]$ MDS code over $\mathbf{F}_{q^2}$.

We need to check the exponential sum $\Sigma _{j \in \mathbf{M}_1} \theta ^{jm_1(q+1+t_1+t_2q)}+\Sigma _{j \in \mathbf{M}_2}$ $\theta ^{jm_2(q+1+t_1+t_2q)}$ for the purpose to get the Hermitian self-orthogonal codes.

Theorem 2.3

Let $m_1, m_2, m_3, m_4, M$ and w be positive integers as above. If for all non-negative integers $t_1$ and $t_2$ satisfying $0 \le t_1, t_2 \le w-1$, $q+1+t_1+t_2q$ cannot be divisible by $m_3$ and $m_4$, then the code $\mathbf{C_M}$ is Hermitian self-orthogonal. When $w < \frac{k_2+1}{2k_2+1}(q-1)$, the above condition is satisfied.

Proof

The conclusion follows from the proof of Theorem 2.1 and the fact $w < min\{\frac{k_1+1}{2k_1+1}(q-1), \frac{k_2+1}{2k_2+1}(q-1)\}$. $\square$

Corollary 2.2

Suppose that q is an even prime power $2^h$, $m_1=2k_1+1$ and $m_2=2k_2+1$ are odd positive integers satisfying $\gcd (m_1,m_2)=1, m_1<m_2$ and $m_1|q+1, m_2|q+1$. We set $m_3=\frac{q^2-1}{m_1}$, $m_4=\frac{q^2-1}{m_2}$, $M=m_3+m_4-\frac{2(q^2-1)}{m_1m_2}$. For each positive integer d in the range $2 \le d \le \lfloor \frac{k_2+1}{2k_2+1}(q-1)+1 \rfloor$, there is a length M q-ary quantum MDS code with minimum distance d.

From Lemma 2.2 we can generalize our recent results to the case that $q+1$ has several factors $m_1,\ldots m_s$, where $\gcd (m_{s_1}, m_{s_2})=1$ for $s_1 \ne s_2$. Some quantum MDS codes coming from Corollary 2.2 are listed in Table 2.

Table 2 $[[(m_1+m_2-2)(2^h-1), (m_1+m_2-2)(2^h-1)-2k, k+1]]_{2^h}$ quantum MDS codes

Actually in the case q is an odd prime power we can use equivalent codes to get new quantum MDS codes as follows. If q is an odd prime power, then 2 is a nonzero element in $\mathbf{F}_q \subset \mathbf{F}_{q^2}$. If $m_1=2k_1+1<m_2=2k_2+1$ are two odd factors of $q+1$, then we have the following identity. When t is not divisible by $\frac{q^2-1}{m_1}$ or $\frac{q^2-1}{m_2}$,

$$\begin{aligned} \Sigma _{j=1}^{\frac{q^2-1}{m_1}} \theta ^{m_1tj}+\Sigma _{j=1}^{\frac{q^2-1}{m_2}} \theta ^{m_2jt}=0 \end{aligned}$$

For those indices j’s which are in both summands, that is, $j=m_1m_2j^{\prime }$, we have $2 \theta ^{m_1m_2 tj^{\prime }}$ in the above identity. Since $2=u^{q+1}$ for some $u \in \mathbf{F}_{q^2}$, the equivalent code can be used to get a Hermitian orthogonal code from Lemma 1.2. Hence we have the following result.

Theorem 2.4

Suppose that q is an odd prime power, $m_1=2k_1+1$ and $m_2=2k_2+1$ are odd positive integers satisfying $\gcd (m_1,m_2)=1, m_1<m_2$ and $m_1|q+1, m_2|q+1$. We set $m_3=\frac{q^2-1}{m_1}$, $m_4=\frac{q^2-1}{m_2}$, $M=m_3+m_4-\frac{q^2-1}{m_1m_2}$. For each positive integer d in the range $2 \le d \le \lfloor \frac{k_2+1}{2k_2+1}(q-1)+1\rfloor$, there is a length M q-ary quantum MDS code with minimum distance d.

In Table 3 we give some new quantum MDS q-ary codes from Theorem 2.4.

Table 3 $[[\frac{q^2-1}{m_1}+\frac{q^2-1}{m_2}-\frac{q^2-1}{m_1m_2}, \frac{q^2-1}{m_1}+\frac{q^2-1}{m_2}-\frac{q^2-1}{m_1m_2}-2k, k+1]]_{q}$ quantum MDS codes

3 New quantum codes II

3.1 Odd q and even $m|q-1$ (Recovery of Theorem 4.11 in [3])

Suppose q is an odd prime power and $q-1=2^ha_1a_2$ where $a_1$ and $a_2$ are odd numbers. We assume $m=2^{h_1}a_1 \ge 6$ is an even factor of $q-1$ where $h_1 \le h$. We first prove the following lemma.

Lemma 3.1

When $0 \le t_1, t_2 \le \frac{q+1}{2}+2^{h-h_1}a_2-2$, the following equality holds:

$$\begin{aligned} \Sigma _{j=1}^{\frac{q^2-1}{m}} \theta ^{jm(t_1+t_2q+\frac{q+1}{2})}=0 \end{aligned}$$

Proof

From the condition $m \ge 6$, $t_1+t_2q+\frac{q+1}{2}<q^2-1$. Thus if $(t_1+\frac{q+1}{2})+t_2q$ is divisible by $\frac{q^2-1}{m}$, the quotient $u < m$. In the case $t_1+\frac{q+1}{2} \le q-1$ we have $u\frac{q^2-1}{m}=t_2q+t_1+\frac{q+1}{2}$. The quotient is $t_2$ and the remainder is $t_1+\frac{q+1}{2}$. The quotient and the remainder have to be the same since $u(\frac{q-1}{m})$ is an integer.

Table 4 $[[(m_1+m_2-1)(2m_1m_2+2), (m_1+m_2-1)(2m_1m_2+2)-2d+2, d]]_{q}$ quantum MDS codes

Since $t_1+\frac{q+1}{2}=t_2$ is divisible by $\frac{q-1}{m}$, $t_1+1+\frac{q-1}{2}$ is divisible by $\frac{q-1}{m}=2^{h-h_1}a_2$. From $t_1 \ge 0$ we have $t_1+1 \ge 1$, and $t_1 \ge 2^{h-h_1}a_2-1$. On the other hand $t_2=t_1+\frac{q+1}{2}$, $t_2 \ge \frac{q+1}{2}+2^{h-h_1}a_2-1$. This is a contradiction. Thus $t_1+t_2q+\frac{q^2-1}{2^{h-h_1+1}m}$ is not divisible by $\frac{q^2-1}{m}$.

In the case $t_1+\frac{q+1}{2} \ge q$ we have $u\frac{q^2-1}{m}=(t_2+1)q+(t_1-\frac{q-1}{2})$. The quotient is $t_2+1$ and the remainder is $t_1-\frac{q-1}{2}$. These two numbers have to be the same since $u< m$. Thus $t_2+1=t_1-\frac{q-1}{2}$ is divisible by $\frac{q-1}{m}=2^{h-h_1}a_2$. From $t_2+1 \ge 1$, we have $t_2 \ge 2^{h-h_1}a_2-1$. Thus $t_1 \ge t_2+1+\frac{q-1}{2} \ge \frac{q+1}{2}+2^{h-h_1}a_2-1$. This is a contradiction.

The code is the set $\{(f(\theta ^{ml}), f(\theta ^{2ml}),\ldots ,f(\theta ^{jml}),\ldots ,f(\theta ^{\frac{q^2-1}{m}ml}): \deg (f) <k\}$. In Lemma 1.2 we can set $v_j'=\theta ^{j\frac{m(q+1)}{2}} \in \mathbf{F}_q^{*}$. $\mathbf{g}_l=(\theta ^{ml}, \theta ^{2ml},\ldots ,\theta ^{jml}, \ldots ,\theta ^{\frac{q^2-1}{m}ml})$, where $0 \le l \le k-1$. Thus a $[\frac{q^2-1}{m}, k]_{q^2}$ Hermitian self-orthogonal MDS code can be constructed from Lemmas 1.2 and 3.1, where k is in the range $1 \le k \le \frac{q+1}{2}+2^{h-h_1}a_2-1$. From Theorem 1.1 we have a length $\frac{q^2-1}{m}$ quantum MDS q-ary code with the minimum distance $d=k+1$ in the range $2 \le d \le \frac{q+1}{2}+2^{h-h_1}a_2$. $\square$

Theorem 3.1

If $q=2^ha_1a_2+1$ is an odd prime power where $a_1$ and $a_2$ are odd numbers and $m=2^{h_1}a_1 \ge 6$ is an even factor of $q-1$ where $h_1 \le h$, then for each integer d in the range $2 \le d \le \frac{q+1}{2}+2^{h-h_1}a_2$, we have a q-ary quantum MDS code with length $\frac{q^2-1}{m}$ and minimum distance d.

This recovers Theorem 4.11 in [3].

3.2 Length $\frac{w(q^2-1)}{u}$ quantum q-ary MDS codes

The main idea of the construction in this subsection is similar to the Sect. 2.2. We add some identities in Lemma 3.1 to get some new identities that some exponential sums are zero. Thus we can construct some new Hermitian self-orthogonal codes.

Suppose $m_1=2^{h_1}a_1 \ge 6$ and $m_2=2^{h_2}b_1 \ge 6$ are two even factors of $q-1=2^ha_1a_2=2^hb_1b_2$ where $a_1,a_2,b_1,b_2$ are odd numbers. Then we have two identities from Lemma 3.1. The addition of these two identities gives another identity. For those indices j which are divisible by both $m_1$ and $m_2$, we have to use the element $\theta ^{j \frac{m_1(q+1)}{2}}+\theta ^{j\frac{m_2(q+1)}{2}} \in \mathbf{F}_q$. It is obvious that this is a nonzero element in $\mathbf{F}_q^{*}$ when $lcm(m_1,m_2)=q-1$ (here lcm is the least common multiple). Set $\mathbf{M}_1$ the set of indices $m_1 \cdot \{1,\ldots ,\frac{q^2-1}{m_1}\}$ and $\mathbf{M}_2=m_2 \cdot \{1,\ldots ,\frac{q^2-1}{m_2}\}$, $\mathbf{M}=\mathbf{M}_1 \cup \mathbf{M}_2$. Here $|\mathbf{M}|=|\mathbf{M}_1|+|\mathbf{M}_2|-\frac{q-1}{lcm(m_1,m_2)}(q+1)=\frac{q^2-1}{m_1}+\frac{q^2-1}{m_2}-(q+1)$ when $lcm(m_1, m_2)=q-1$. The code is the set $\{(f(x))_{x \in \mathbf{M}}: 0 \le \deg (f) \le k-1\}$, where $1\le k \le \frac{q-1}{2}+\min \{2^{h-h_1}a_2,2^{h-h_2}b_2\}$.

Theorem 3.2

Assuming that $q=2^{h_1}a_1a_2+1=2^{h_2}b_1b_2+1$ is an odd prime power as above and $a_1, a_2, b_1, b_2$ are odd numbers. Suppose also that $m_1=2^{h_1}a_1$ and $m_2=2^{h_2}b_1$ are two even factors of $q-1$ satisfying $lcm(m_1,m_2)=q-1$ as above. Then for each integer d in the range $2 \le d \le \frac{q+1}{2}+min\{2^{h-h_1}a_2, 2^{h-h_2}b_2\}$ we have a q-ary quantum MDS code with length $|\mathbf{M}|=\frac{q^2-1}{m_1}+\frac{q^2-1}{m_2}-(q+1)$ and minimum distance d.

Corollary 3.1

If $2m_1m_2+1$ is a prime power where $m_1<m_2$ are two co-prime odd numbers, then for each integer d in the range $2 \le d \le m_1m_2+m_1+1$ we have a length $\frac{(m_1+m_2-1)(q^2-1)}{2m_1m_2}=(m_1+m_2-1)(2m_1m_2+2)$ q-ary quantum MDS code and the minimum distance d.

We list some new quantum MDS codes from Corollary 3.1 in Table 4.

4 New quantum codes III

Just as in Sect. 2.2 the idea of the construction in this section is that the addition of two identities in Lemmas 1.1 and 3.1 gives us some new identities showing that some exponential sums are zero. This leads to some new Hermitian self-orthogonal codes with different lengths.

Let q be an odd prime power and $m_1=2k_1+1$ is an odd factor of $q+1$. From Theorem 2.1 we have that the following identity holds when $0 \le t_1,t_2 \le \frac{q-1}{2}+\frac{q+1}{2m_1}-2$.

$$\begin{aligned} \Sigma _{j=1}^{\frac{q^2-1}{m_1}} \theta ^{jm_1(t_1+t_2q)} \cdot \theta ^{jm_1(q+1)}=0 \end{aligned}$$

From Lemma 3.1 if $m_2|q-1$ is an even factor of $q-1$ we have the following identity when $0 \le t_1, t_2 \le \frac{q-1}{2}+\frac{q-1}{m_2}-1$.

$$\begin{aligned} \Sigma _{j=1}^{\frac{q^2-1}{m_2}} \theta ^{jm_2(t_1+t_2q)} \cdot \theta ^{j\frac{m_2(q+1)}{2}}=0 \end{aligned}$$

We can get the following identity by adding these two identities.

$$\begin{aligned} \Sigma _{j=1}^{\frac{q^2-1}{m_1}} \theta ^{jm_1(t_1+t_2q)} \cdot \theta ^{jm_1(q+1)}+ H(\Sigma _{j=1}^{\frac{q^2-1}{m_2}} \theta ^{jm_2(t_1+t_2q)} \cdot \theta ^{j\frac{m_2(q+1)}{2}})=0 \end{aligned}$$

Here H can be any nonzero $H \in \mathbf{F}_q^{*}$ and the common $t_1$ and $t_2$ are in the range $0 \le t_1, t_2 \le \frac{q-1}{2}+\ min \{\frac{q+1}{2m_1}-2, \frac{q-1}{m_2}-1\}$. At the position $\theta ^{m_1m_2t}$ it is clear that $\theta ^{m_1^2m_2t(q+1)}+H\theta ^{\frac{m_1m_2^2t(q+1)}{2}}$ is an element in $\mathbf{F}_q$. Since $\theta ^{(m_1-\frac{m_2}{2})m_1m_2t(q+1)}$ can only be the $\frac{q-1}{m_2}$ nonzero elements in the subgroup of $\mathbf{F}_q^{*}$ generated by $\theta ^{m_2(q+1)}$, there exists a $H \in \mathbf{F}_q^{*}$ such that $\theta ^{m_1^2m_2t(q+1)}+H\theta ^{\frac{m_1m_2^2t(q+1)}{2}}$ is a nonzero element in $\mathbf{F}_q^{*}$ for any possible t.

Let $\mathbf{M}$ be the set $\{\theta ^{jm_1}: j=1,\ldots ,\frac{q^2-1}{m_1}\} \cup \{\theta ^{jm_2}: j=1,\ldots ,\frac{q^2-1}{m_2}\}$. The code is the set $\{(f(x))_{x \in \mathbf{M}}: 0 \le \deg (f) \le \frac{q-1}{2}+\min \{\frac{q+1}{2m_1}-2, \frac{q-1}{m_2}-1\}\}$. This is equivalent to a Reed–Solomon code.

Theorem 4.1

If q is an odd prime power, $m_1$ is an odd factor of $q+1$ and $m_2$ an even factor of $q-1$, then for each integer d in the range $2 \le d \le \frac{q-1}{2}+\min \{\frac{q+1}{2m_1}, \frac{q-1}{m_2}+1\}$, we have a q-ary quantum MDS code with length $\frac{q^2-1}{m_1}+\frac{q^2-1}{m_2}-\frac{q^2-1}{m_1m_2}$ and minimum distance d.

Actually Theorem 4.1 is quite general as illustrated in the following results.

Corollary 4.1

Let q be an odd prime power. If there exists an odd integer $m | q+1$ such that $m-1$ is an even factor of $q-1$. Then for each integer d in the range $2 \le d \le \frac{q-1}{2}+\frac{q+1}{2m}$ we have a length $\frac{2(q^2-1)}{m}$ q-ary quantum MDS code with minimum distance d.

There are many such odd prime powers q as illustrated in Table 5.

Table 5 Quantum MDS codes with lengths $\frac{2(q^2-1)}{m}$

The lengths of some quantum MDS q-ary codes in Table 5 have the form $4(q-1)$ where q is an odd prime power such that $(q+1)$ is not divisible by 4. This case is not covered in the previous results (see the table in page 1482 of [3]).

Corollary 4.2

If q is an odd prime power of the form $q \equiv 1$ mod 4, then for each integer d in the range $2 \le d \le \frac{q+1}{2}$ we have a length $4(q-1)$ q-ary quantum MDS code with minimum distance d.

From the main result in [9] (or see 3 in the table in page 1482 of [3]), only the range $3 \le d \le \frac{q-1}{2}$ is allowed. Our result gives a quantum q-ary MDS $[[4(q-1), 3q-3, \frac{q+1}{2}]]_{q}$ code when $q=4k+1$ is an odd prime power.

Corollary 4.3

Let q be an odd prime power. If there exists an even factor $2(2k+1)$ of $q-1$ such that $4k+1$ is a odd factor of $q+1$, then for each integer d in the range $2 \le d \le \frac{q-1}{2}+\frac{q+1}{2(4k+1)}$ we have a length $\frac{q-1}{2k+1} \cdot (q+1)$ q-ary quantum MDS code with minimum distance d.

In Theorem 4.11 of [3] and Theorem 3.1 here m cannot be an odd factor. This Corollary 4.3 partially solves this case under an assumption on q. However, there are a lot of such odd prime powers q and odd factors $(2k+1)|q-1$ as illustrated in Table 6.

Table 6 Quantum MDS codes with lengths $\frac{(q-1)}{2k+1} \cdot (q+1)$

5 New quantum codes IV

In this section we treat the case that q is an odd prime power and $n=\frac{q^2-1}{m}$, where $m|q^2-1$, and m is neither a factor of $q-1$ nor $q+1$.

We need the following two lemmas.

Lemma 5.1

If $m_1$ is an even integer and $m_2$ is an odd integer satisfying $\gcd (m_1,m_2)=1$, there are infinitely many primes q satisfying $m_1|q-1$ and $m_2|q+1$.

Proof

Since $gcd(m_1, m_2)=1$ we have two integers $l_0$ and $k_0$ satisfying $l_0m_1+2=k_0m_2$. Thus $l=l_0+m_2t$ and $k=k_0+m_1t$ also satisfy $lm_1+2=km_2$ for all integers $t=0 \pm 1, \pm 2, \ldots$. It is clear $\gcd (l_0m_1+1,m_1)=1$. We have $l_0m_1+1+1=k_0m_2$, then $\gcd (l_0m_1+1,m_2)=1$.

From Dirichlet Theorem there are infinitely many primes in the arithmetic sequence $m_2m_1t+l_0m_1+1$ because of $\gcd (l_0m_1+1,m_1m_2)=1$. It is direct to verify $m_1|q-1$ and $m_2|q+1$. $\square$

Lemma 5.2

There are infinitely many pairs of positive integers $(m_1,m_2)$ satisfying the following conditions.

• 1) $m_1$ is even, $m_2$ is odd and $\gcd (m_1,m_2)=1$;

• 2) $\frac{m_1+m_2-1}{m_1m_2}=\frac{1}{m}$ where m is a positive integer satisfying $gcd(m_1,m) >1$ and $gcd(m_2,m) >1$.

Proof

We consider $m_2=k_1k_2$ where $k_1$ and $k_2$ are odd numbers. Set $k_3$ and $k_4$ two un-determined positive integers satisfying $k_1k_2-1+2k_3k_4=k_1k_3$. Then $k_1k_2-1=k_3(k_1-2k_4)$. From the factorization of $k_1k_2-1$ we get suitable $k_3$ and $k_4$. Hence $m_1=k_1k_2$ and $m_2=2k_3k_4$ are the integers satisfying the conditions.

For example when $k_1=35$ and $k_2=3$, $105-1=8 \cdot 13=k_3(35-2k_4)$, we can set $k_3=8$ and $k_4=11$. Then $m_1=176$ and $m_2=105$. $\frac{105+176-1}{176 \cdot 105}=\frac{1}{66}$. When $k_1=35$ and $k_2=5$. $174=6 \cdot 29=k_3(35-2k_4)$, we can set $k_3=6$ and $k_4=3$. Then $m_1=36$ and $m_2=175$. $\frac{175+36-1}{36 \cdot 175}=\frac{1}{30}$. $\square$

Theorem 5.1

There are infinitely many pairs of integers $(m_1,m_2)$ as in Lemma 5.2 and infinitely many primes q as in Lemma 5.1 for each such pair $(m_1, m_2)$. For each such pair $(m_1,m_2)$ and the infinitely many primes q as in Lemma 5.1, we have a q-ary quantum MDS code with length $n=\frac{q^2-1}{m}$ and minimum distance d for each integer d in the range $2 \le d \le \frac{q-1}{2}+\min \{\frac{q+1}{2m_2}, \frac{q-1}{m_1}+1\}$.

Proof

The conclusion follows from Lemmas 5.1 and 5.2 and Theorem 4.1 directly.

We list some new q-ary quantum MDS codes from Theorem 5.1 in Table 7.

Table 7 Quantum MDS codes from Theorem 5.1

$\square$

Corollary 5.1

Let k be any positive integer satisfying $k \equiv 5$ mod 9. If $q=16k^2-12k+1$ is an odd prime power, then we have a q-ary quantum MDS code with length $\frac{q^2-1}{3k}$ and minimum distance d for each integer d in the range $2 \le d \le \frac{q+1}{2}+\frac{2k-1}{3}$.

Proof

Set $m_1=4k$ and $m_2=3(4k-1)$ in Theorem 5.1 we get the conclusion.

For example when $k=14$ and $q=2969$ is a prime we have a 2969-ary quantum MDS $[[209880, 209880-2d+2, d]]_{2969}$ code for each integer d in the range $2 \le d \le 1494$. In the above Corollary 5.1 we should note that 3k is not a factor of $q-1$ or $q+1$. This case has not been treated in the previous works [3, 8–13]. $\square$

6 Summary

In this paper we give a direct method constructing $q^2$-ary Hermitian self-orthogonal MDS codes with dimensions $k> \frac{q}{2}$. This leads to many new q-ary quantum MDS codes with minimum distances $d>\frac{q}{2}$. Some new q-ary quantum MDS codes with $q> \frac{q}{2}$ constructed in this paper are listed in Table 8.

Table 8 Quantum MDS codes