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arXiv:1709.02580

ttt-Frobenius Negacyclic Codes

Priyabrata Bag, Dey, Santanu

Published Sep 8, 2017• 0 citations
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Abstract

Let $\mathbb{F}_p$ denote the finite field of order $p$, where $p$ is an odd prime. We study certain quantum negacyclic codes over $\mathbb{F}_p$ which we call $t$-Frobenius negacyclic codes. We obtain a criterion for constructing such codes from certain subspaces of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ in terms of generating pairs of ideals of cyclotomic rings and we completely classify all linear $t$-Frobenius negacyclic codes. Further, the notion of BCH distance is extended for these codes and several new codes are identified.

DOI
10.48550/arxiv.1709.02580

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Let Fp\mathbb{F}_pFp​ denote the finite field of order ppp, where ppp is an odd prime. We study certain quantum negacyclic codes over Fp\mathbb{F}_pFp​ which we call ttt-Frobenius negacyclic codes. We obtain a criterion for constructing such codes from certain subspaces of Fpn×Fpn\mathbb{F}_p^n\times\mathbb{F}_p^nFpn​×Fpn​ in terms of generating pairs of ideals of cyclotomic rings and we completely classify all linear ttt-Frobenius negacyclic codes. Further, the notion of BCH distance is extended for these codes and several new codes are identified.
Let [Formula: see text] be an odd prime power and [Formula: see text] be a positive integer. Maximum designed distance such that negacyclic BCH codes over [Formula: see text] of length [Formula: see text] are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length [Formula: see text] are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes.
In this paper, the properties of q2-ary cyclotomic cosets modulo 2n are investigated in detail, where n=[(q2m-1)/(m(q-1))],q ≡ ±1mod m or q2≡ -1mod m when m is even and q is an odd power prime when m is power of 2. Then, non-narrow-sense negacyclic Bose-Chaudhuri-Hocquenghem (BCH) codes of length n that contain their Hermitian duals are determined. By the Hermitian construction, series of quantum codes are obtained from such negacyclic BCH codes. Compared with quantum codes in the [6], [10], [12], our constructed ones have better parameters.
Quantum Cyclic Code of length dividing pt+1p^{t}+1pt+1
In this paper, we study cyclic stabiliser codes over Fp\mathbb{F}_pFp​ of length dividing pt+1p^t+1pt+1 for some positive integer ttt. We call these ttt-Frobenius codes or just Frobenius codes for short. We give methods to construct them and show that they have efficient decoding algorithms. An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We completely characterise all linear Frobenius codes. As a consequence, we show that for every integer nnn that divides pt+1p^t+1pt+1 for an odd ttt, there are no linear cyclic codes of length nnn. On the other hand for even ttt, we give an explicit method to construct all of them. This gives us a many explicit example of Frobenius codes which include the well studied Laflamme code. We show that the classical notion of BCH distance can be generalised to all the Frobenius codes that we construct, including the non-linear ones, and show that the algorithm of Berlekamp can be generalised to correct quantum errors within the BCH limit. This gives, for the first time, a family of codes that are neither CSS nor linear for which efficient decoding algorithm exits. The explicit examples that we construct are summarised in Table \ref{tab:explicit-examples-short} and explained in detail in Tables \ref{tab:explicit-examples-2} (linear case) and \ref{tab:explicit-examples-3} (non-linear case).
In this paper, we study cyclic stabiliser codes over Fpof length dividing pt+ 1 for some positive integer t. We call these t-Frobenius codes or just Frobenius codes for short. We give methods to construct them and show that they have efficient decoding algorithms. An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We completely characterise all linear Frobenius codes. As a consequence, we show that for every integer n that divides pt+ 1 for an odd t, there are no linear cyclic codes of length n. On the other hand for even t, we give an explicit method to construct all of them. This gives us many explicit examples of Frobenius code which include the well studied Laflamme code. We show that the classical notion of BCH distance can be generalised to all the Frobenius codes that we construct, including the non-linear ones, and show that the algorithm of Berlekamp can be generalised to correct quantum errors within the BCH limit. This gives, for the first time, a family of codes that are neither CSS nor linear for which efficient decoding algorithm exits.
In this paper, we construct families of nonbinary quantum codes from classical negacyclic codes. The BCH bound for negacyclic codes is applied to determine the dimension and distance of these quantum codes. The parameters of these quantum codes are good compared with the ones available in the literature. We also construct a family of quantum maximum-distance-separable codes with parameters ${[[q+1,q\ensuremath{-}2d+3,d]]}_{q}$, where $q\ensuremath{\equiv}1\phantom{\rule{0.28em}{0ex}}\mathrm{mod}\phantom{\rule{0.28em}{0ex}}4$ and $2\ensuremath{\le}d\ensuremath{\le}(q+3)/2$.