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Register a new userSome New Constructions of Quantum MDS Codes
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Added by
Weijun Fang
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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.
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Maximum distance separable (MDS) codes are optimal where the minimum distance cannot be improved for a given length and code size. Twisted Reed-Solomon codes over finite fields were introduced in 2017, which are generalization of Reed-Solomon codes. Twisted Reed-Solomon codes can be applied in cryptography which prefer the codes with large minimum distance. MDS codes can be constructed from twisted Reed-Solomon codes, and most of them are not equivalent to Reed-Solomon codes. In this paper, we first generalize twisted Reed-Solomon codes to generalized twisted Reed-Solomon codes, then we give some new explicit constructions of MDS (generalized) twisted Reed-Solomon codes. In some cases, our constructions can get MDS codes with the length longer than the constructions of previous works. Linear complementary dual (LCD) codes are linear codes that intersect with their duals trivially. LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. We also provide new constructions of LCD MDS codes from generalized twisted Reed-Solomon codes.
The parameters of MDS self-dual codes are completely determined by the code length. In this paper, we utilize generalized Reed-Solomon (GRS) codes and extended GRS codes to construct MDS self-dual (self-orthogonal) codes and MDS almost self-dual codes over. The main idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual (self-orthogonal). The evaluation sets are consists of two subsets which satisfy some certain conditions and the length of these codes can be expressed as a linear combination of two factors of q-1. Four families of MDS self-dual codes, two families of MDS self-orthogonal codes and two families of MDS almost self-dual codes are obtained and they have new parameters.
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.
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.
Systematic constructions of MDS self-dual codes is widely concerned. In this paper, we consider the constructions of MDS Euclidean self-dual codes from short length. Indeed, the exact constructions of MDS Euclidean self-dual codes from short length ($n=3,4,5,6$) are given. In general, we construct more new of $q$-ary MDS Euclidean self-dual codes from MDS self-dual codes of known length via generalized Reed-Solomon (GRS for short) codes and extended GRS codes.
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