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Quantum variations of cyclotomic cosets for cyclic stabiliser codes construction

En Chong Yap, Kai Lin Ong

Published Jun 17, 2024• 2 citations
Publisher

Abstract

Abstract Cyclotomic cosets are intrinsically linked with the design and construction of classical cyclic codes whose properties can be inferred from the coset structures. This paper proposes some new quantum variations of cyclotomic cosets for cyclic stabiliser construction. These variations are governed by several parameters which are devoted to designing the two essential parts: the error part and the position part of these cosets. Criteria on these cosets in generating additive cyclic stabiliser are extensively studied, followed by actual implementation on stabiliser codes construction of several selected classes of length.

DOI
10.1007/s11128-024-04451-5

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Abstract Cyclotomic cosets are intrinsically linked with the design and construction of classical cyclic codes whose properties can be inferred from the coset structures. This paper proposes some new quantum variations of cyclotomic cosets for cyclic stabiliser construction. These variations are governed by several parameters which are devoted to designing the two essential parts: the error part and the position part of these cosets. Criteria on these cosets in generating additive cyclic stabiliser are extensively studied, followed by actual implementation on stabiliser codes construction of several selected classes of length.
In this paper, we use q -cyclotomic cosets modulo n = q − 1 and generator polynomial to describe repeated-root cyclic codes of length N = p α n . We determine the condition under which a repeated-root cyclic code contains its dual code. Then, for q = 2 a , we give a class of dual-containing repeated-root cyclic codes. Applying CSS construction and Steane construction to these underlying classical codes, many quantum codes with good parameters shall be constructed, which are new in the sense that they are not available in the literature.
A new criterion for the existence of Hermitian dual-containing cyclic codes is obtained based on a characterization of q-cyclotomic cosets modulo n. This criterion admits a method of constructing new Hermitian dual-containing cyclic codes (hence new quantum codes) from the old ones, and enables us to construct several classes of quantum maximum-distance-separable (MDS) codes from classical Hermitian dual-containing cyclic codes. We analyse that our codes are good in certain sense. Besides, some enumerative conclusions are also presented as a result of the characterization of q-cyclotomic cosets modulo n.
In this paper, we define and study \emph{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]][[5,1,3]][[5,1,3]], [[17,1,7]][[17,1,7]][[17,1,7]] and [[17,9,3]][[17,9,3]][[17,9,3]], all of which are \emph{not} CSS. The [[5,1,3]][[5,1,3]][[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))((5,6,2))((5,6,2)) nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length 4m+14^m +14m+1 over F2\mathbb{F}_2F2​ 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 $\floor{\frac{d-1}{2}}$ errors when the BCH distance is ddd.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields have been studied extensively. One can construct quantum stabilizer codes with good parameters using classical BCH codes. In this paper, our goal is to find such classical BCH codes. We study some properties of suitable cyclotomic cosets at first. These results make it possible to construct nonbinary quantum BCH codes with a given parameter set. Several new families of quantum BCH codes obtained are based on Steane's enlargement of nonbinary Calderbank-Shor-Steane codes and Hermitian construction, respectively. Meanwhile, we have shown that the cyclotomic cosets given in our schemes are optimal to design quantum BCH codes. The defining set contains most consecutive integers. Therefore, corresponding quantum stabilizer codes have better lower bound of minimum distance. Furthermore, it is convenient to compute the dimension of new quantum codes. Compared with the ones available in the literature, the quantum BCH codes in our schemes have good parameters. In particular, we extend known results to more general case.
In order to constructing quantum BCH codes from classical BCH codes using CSS constructing method it needs to determine whether the BCH codes contains its dual. Cyclotomic cosets play very important roles in classical error correction theory and quantum error correction theory. This letter presented several properties of cyclotomic cosets and introduced a method to determine whether a non-primitive BCH code over GF (q) contains its dual. The method presented in this paper with polynomial time complexity can also be applied to nonnarrow sense primitive BCH codes.
In this paper, we use 2-cyclotomic cosets of modulo n and generator polynomials to describe binary cyclic codes of length N=2αn with n odd. We discuss the conditions under which two cyclic codes [Formula: see text] and [Formula: see text] can be used to construct quantum codes by CSS construction or Steane's construction. Using the results of Chen, Promhouse and Tavares, and Castagnoli et al., we study the quantum codes that can be constructed from binary cyclic codes of length N=2αn with n odd and n≤99, and α≤2. We find that except the quantum codes constructed by Steane, there are also some very interesting quantum codes constructed from repeated-root cyclic codes, and some of the quantum codes constructed by Steane can be improved.