2010/07/10 by Sagarmoy Dutta, Dutta, Sagarmoy, Piyush P Kurur +2 · 1 citation
Computer Science · Mathematics · #FOS: Computer and information sciences #Information Theory (cs.IT) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum-Dot Cellular Automata #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1007.1697
arxiv created 2010/07/10 · openalex publication_date 2010/07/10 · arxiv updated 2010/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we define and study quantum cyclic codes, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters [[5,1,3]], [[17,1,7]] and [[17,9,3]], all of which are not CSS. The [[5,1,3]] code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the ((5,6,2)) nonstabiliser first constructed by Rains\etal~ citerains97nonadditive and latter by Arvind \etal~\citearvind:2004:nonstabilizer is cyclic. We also study stabiliser codes of length 4m +1 over \mathbbF2 for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to \floor(d-1)/(2) errors when the BCH distance is d.