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Register a new userNew Quantum MDS Codes over Finite Fields
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Added by
Xiaolei Fang
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, we present three new classes of $q$-ary quantum MDS codes utilizing generalized Reed-Solomon codes satisfying Hermitian self-orthogonal property. Among our constructions, the minimum distance of some $q$-ary quantum MDS codes can be bigger than $frac{q}{2}+1$. Comparing to previous known constructions, the lengths of codes in our constructions are more flexible.
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In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power $q$ with $q$ square, the total number of lengths for MDS self-dual codes over $mathbb{F}_q$ presented in this paper is much more than those in all the previous results.
In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $mathbb{F}_3$ and $mathbb{F}_4$. We study the bound of LCD codes over $mathbb{F}_q$.
It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than q/2+1 Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in [1] and [2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.
Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters [ [[tq, tq-2d+2, d]]_{q} ] for any $1 leq t leq q, 2 leq d leq lfloor frac{tq+q-1}{q+1}rfloor+1$, and [ [[t(q+1)+2, t(q+1)-2d+4, d]]_{q} ] for any $1 leq t leq q-1, 2 leq d leq t+2$ with $(p,t,d) eq (2, q-1, q)$. Our quantum codes have flexible parameters, and have minimum distances larger than $frac{q}{2}+1$ when $t > frac{q}{2}$. Furthermore, it turns out that our constructions generalize and improve some previous results.
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.
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