# 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

, Volume 15 (7) – Apr 18, 2016
14 pages

/lp/springer_journal/new-q-ary-quantum-mds-codes-with-distances-bigger-than-frac-q-2-q-2-ywE8Lgs5Dj
Publisher
Springer US
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

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