# A class of constacyclic BCH codes and new quantum codes

A class of constacyclic BCH codes and new quantum codes Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length $$n=q^{2m}+1$$ n = q 2 m + 1 over $$F_{q^2}$$ F q 2 with q an odd prime power, there are only the ones of distance $$\delta \le 2q^2$$ δ ≤ 2 q 2 are obtained in the literature. In this paper, by a detailed analysis of properties of $$q^{2}$$ q 2 -ary cyclotomic cosets, maximum designed distance $$\delta _\mathrm{{max}}$$ δ max of a class of Hermitian dual-containing constacyclic BCH codes with length $$n=q^{2m}+1$$ n = q 2 m + 1 are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over $$F_{q^2}$$ F q 2 . Then we can obtain a sequence of dual-containing constacyclic codes of designed distances $$2q^2<\delta \le \delta _\mathrm{{max}}$$ 2 q 2 < δ ≤ δ max . Consequently, new quantum codes with distance $$d > 2q^2$$ d > 2 q 2 can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# A class of constacyclic BCH codes and new quantum codes

, Volume 16 (3) – Feb 2, 2017
16 pages

/lp/springer_journal/a-class-of-constacyclic-bch-codes-and-new-quantum-codes-Gm6GAVnnUW
Publisher
Springer Journals
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/s11128-017-1533-y
Publisher site
See Article on Publisher Site

### Abstract

Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length $$n=q^{2m}+1$$ n = q 2 m + 1 over $$F_{q^2}$$ F q 2 with q an odd prime power, there are only the ones of distance $$\delta \le 2q^2$$ δ ≤ 2 q 2 are obtained in the literature. In this paper, by a detailed analysis of properties of $$q^{2}$$ q 2 -ary cyclotomic cosets, maximum designed distance $$\delta _\mathrm{{max}}$$ δ max of a class of Hermitian dual-containing constacyclic BCH codes with length $$n=q^{2m}+1$$ n = q 2 m + 1 are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over $$F_{q^2}$$ F q 2 . Then we can obtain a sequence of dual-containing constacyclic codes of designed distances $$2q^2<\delta \le \delta _\mathrm{{max}}$$ 2 q 2 < δ ≤ δ max . Consequently, new quantum codes with distance $$d > 2q^2$$ d > 2 q 2 can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Feb 2, 2017

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