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

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 - 2 d + 2 , d ] ] q codes with minimum distances $$d>\frac{q}{2}$$ d > q 2 for sparse lengths $$n>q+1$$ n > q + 1 . In the case $$n=\frac{q^2-1}{m}$$ n = q 2 - 1 m 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 = q 2 - 1 m while $$m|q^2-1$$ m | q 2 - 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 > q 2 has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over $$\mathbf{F}_{q^2}$$ 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 $$\frac{w(q^2-1)}{u}$$ w ( q 2 - 1 ) u and minimum distances $$d > \frac{q}{2}$$ d > q 2 are presented. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-016-1311-2
Publisher site
See Article on Publisher Site

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 $$[[n,n-2d+2,d]]_q$$ [ [ n , n - 2 d + 2 , d ] ] q codes with minimum distances $$d>\frac{q}{2}$$ d > q 2 for sparse lengths $$n>q+1$$ n > q + 1 . In the case $$n=\frac{q^2-1}{m}$$ n = q 2 - 1 m 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 = q 2 - 1 m while $$m|q^2-1$$ m | q 2 - 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 > q 2 has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over $$\mathbf{F}_{q^2}$$ 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 $$\frac{w(q^2-1)}{u}$$ w ( q 2 - 1 ) u and minimum distances $$d > \frac{q}{2}$$ d > q 2 are presented.

Journal

Quantum Information ProcessingSpringer Journals

Published: Apr 18, 2016

References

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