Quasi-cyclic Codes Research Articles

On quantum codes derived from quasi-cyclic codes over a non-chain ring

On quantum codes derived from quasi-cyclic codes over a non-chain ring

New distance bounds for quasi-cyclic codes

AbstractWe consider the minimum weight of codewords in a quasi-cyclic code and characterize the estimate in its most general setup using their concatenated structure. The new bound we derive generalizes the Jensen and Güneri–Özbudak bounds and it holds for the more general class of multilevel concatenated codes.

Cyclic Codes over a Non-Commutative Non-Unital Ring

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

Self-orthogonal and quantum codes over chain rings

In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field $\mathbb{F}_q$. We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes.

A provably masked implementation of BIKE Key Encapsulation Mechanism

BIKE is a post-quantum key encapsulation mechanism (KEM) selected for the 4th round of the NIST's standardization campaign. It relies on the hardness of the syndrome decoding problem for quasi-cyclic codes and on the indistinguishability of the public key from a random element, and provides the most competitive performance among round 4 candidates, which makes it relevant for future real-world use cases. Analyzing its side-channel resistance has been highly encouraged by the community and several works have already outlined various side-channel weaknesses and proposed ad-hoc countermeasures. However, in contrast to the well-documented research line on masking lattice-based algorithms, the possibility of generically protecting code-based algorithms by masking has only been marginally studied in a 2016 paper by Chen et al. in SAC 2015. At this stage of the standardization campaign, it is important to assess the possibility of fully masking BIKE scheme and the resulting cost in terms of performances. In this work, we provide the first high-order masked implementation of a code-based algorithm. We had to tackle many issues such as finding proper ways to handle large sparse polynomials, masking the key-generation algorithm or keeping the benefit of the bitslicing. In this paper, we present all the gadgets necessary to provide a fully masked implementation of BIKE, we discuss our different implementation choices and we propose a full proof of masking in the Ishai Sahai and Wagner (Crypto 2003) model. More practically, we also provide an open C-code masked implementation of the key-generation, encapsulation and decapsulation algorithms with extensive benchmarks. While the obtained performance is slower than existing masked lattice-based algorithms, we show that masking at order 1, 2, 3, 4 and 5 implies a performance penalty of x5.8, x14.2, x24.4, x38 and x55.6 compared to order 0 (unmasked and unoptimized BIKE). This scaling is encouraging and no Boolean to Arithmetic conversion has been used.

On the duals of quasi-cyclic codes and their application to quantum codes

On the duals of quasi-cyclic codes and their application to quantum codes

Cyclic Codes over a Non-Local Non-Unital Ring

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Gray Images of Constacyclic Codes over a Non-chain Extension of Z 4

Let [Formula: see text] be the ring of integers modulo 4. We study the [Formula: see text]-constacyclic and [Formula: see text]-cyclic codes over the non-chain ring [Formula: see text] for a unit [Formula: see text] in [Formula: see text]. We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over [Formula: see text] are cyclic, quasi-cyclic or permutation equivalent to this code. For an odd positive integer [Formula: see text], we determine the generator polynomials of cyclic and [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text]. Further, we prove that a [Formula: see text]-cyclic code of length [Formula: see text] is a [Formula: see text]-constacyclic code if [Formula: see text] is odd, and a [Formula: see text]-quasi-twisted code if [Formula: see text] is even. A few examples are also incorporated, in which two parameters are new and one is best known to date.

Constructions of Binary Constant-Weight Quasi-Cyclic Codes and Quasicyclically Permutable Codes

A theorem is proven showing how to obtain a constant-weight binary quasi-cyclic code from a pr-ary linear cyclic code, where p is a prime and r is a positive integer, r > 1, by using a representation of the elements of a Galois field, GF(pr), as cyclic shifts of a binary pr-tuple. From this theorem, constructions are derived for two classes of constant-weight binary quasi-cyclic codes. These two classes are shown to achieve the Johnson upper bound on the number of codewords asymptotically for long block lengths. A quasicyclically permutable (QCP) code is a binary code such that the codewords are quasicyclically distinct and have cyclic order equal to the code block length. A technique is described for selecting virtually the maximum number of cyclically distinct codewords of full cyclic order from Reed-Solomon (RS) codes and from Berlekamp-Justesen (BJ) codes, both known to be maximum distance separable codes. Those cyclically distinct codewords of full cyclic order from RS codes and from BJ codes are mapped to binary to produce two classes of asymptotically optimum constant-weight quasicyclic codes and two classes of asymptotically optimum constant weight QCP codes. An application of QCP codes is introduced to construct protocol-sequence sets for the M-active-out-of-T users collision channel without feedback, allowing more users than strict cyclically permutable codes with the same block length and minimum distance.

Explicit Cyclic and Quasi-Cyclic Codes with Optimal, Best Known Parameters and Large Relative Minimum Distances

Explicit Cyclic and Quasi-Cyclic Codes with Optimal, Best Known Parameters and Large Relative Minimum Distances

On the generalised rank weights of quasi-cyclic codes

<p style='text-indent:20px;'>Generalised rank weights were formulated in analogy to Wei's generalised Hamming weights, but for the rank metric. In this paper we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated, which is tighter than the known Singleton bound. Additionally, it is shown that the first generalised rank weight of self-dual <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-generator quasi-cyclic codes is almost completely determined by the choice of <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb F}_{q^{m}} $\end{document}</tex-math></inline-formula>.</p>

Constacyclic codes over the non-chain finite commutative ring 4 2 2 [ , ] , , u v 〈u − u v uv〉

This paper considers (1+2u) -constacyclic and (1+2u) -skew constacyclic codes over the ring R = Z4[u,v]/<u2 – u, v2, uv>, and some Gray maps over this ring are presented. The Gray images of these codes are shown to be cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes. In addition, (1+2u) constacyclic codes are given with images that are new Z4 codes according to the online database of linear Z4 codes [3].

Quasi-twisted codes as contractions of quasi-cyclic codes

Quasi-twisted codes as contractions of quasi-cyclic codes

Improved Spectral Bound for Quasi-Cyclic Codes

Improved Spectral Bound for Quasi-Cyclic Codes

Fast computation of cyclic convolutions and their applications in code-based asymmetric encryption schemes

The development of fast algorithms for key generation, encryption and decryption not only increases the efficiency of related operations. Such fast algorithms, for example, for asymmetric cryptosystems on quasi-cyclic codes, make it possible to experimentally study the dependence of decoding failure rate on code parameters for small security levels and to extrapolate these results to large values of security levels. In this article, we explore efficient cyclic convolution algorithms, specifically designed, among other things, for use in encoding and decoding algorithms for quasi-cyclic LDPC and MDPC codes. Corresponding convolutions operate on binary vectors, which can be either sparse or dense. The proposed algorithms achieve high speed by compactly storing sparse vectors, using hardware-supported XOR instructions, and replacing modulo operations with specialized loop transformations. These fast algorithms have potential applications not only in cryptography, but also in other areas where convolutions are used.

On generalized quasi-cyclic codes over [formula omitted

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important roles in coding theory. In this paper, we study some results on GQC codes over Z4 including normalized generating sets, minimal generating sets and normalized generating sets of their dual codes. As an application, new Z4-linear codes and good binary nonlinear codes are constructed from GQC codes over Z4.

Some Quaternary Additive Codes Outperform Linear Counterparts

The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codes http://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.

On the structure of monomial codes and their generalizations

In this paper, we are interested in monomial codes with associated vector $a=(a_0, a_1,\ldots, a_{n-1}),$ introduced in \cite{Maria2017}, and more generally in linear codes invariant under a monomial matrix $M=\diag(a_0, a_1,\ldots, a_{n-1}) P_{\sigma}$ where $\sigma$ is a permutation and $P_{\sigma}$ its associated permutation matrix.We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix $M$. Next, we identify monomial codes with associated vector $a=(a_0,a_2,\ldots, a_{n-1})$ by the ideals of the polynomial ring $ R_{_{q,n}}:= \quot{{\Fq[x]}}{{\langle x^{n}-\prod_{i=0}^{n-1}a_i \rangle}},$ via a special isomorphism $\varphi_{_{\overline{a}}}$ which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of $\ell-$quasi-monomial ( $\ell-$QM) codes of length $n=m\ell,$ where on the one hand, we characterize them by the $ R_{_{q,m}}-$submodules of $ R_{_{q,m}}^{\ell}.$ On the other hand, $\ell-$QM codes are seen as additive monomial codes over the extension $\mathbb{F}_{q^{\ell}}/\Fq.$ So, as in the case of quasi-cyclic codes \cite{Guneri2018}, we characterize those codes that have $\mathbb{F}_{q^{\ell}}-$linear images with respect to a basis of the extension $ \mathbb{F}_{q^{\ell}}/\Fq,$ based on the CRT decomposition. Finally, we show that $\ell-$QM codes and additive monomial codes are asymptotically good.

Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds.

On Toeplitz codes of index t and isometry codes

In this paper, we introduce Toeplitz codes of index t, which are a generalization of quasi-cyclic codes of index t. We show that such codes meet the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 1t over Fq, so they are asymptotically good for any t. We characterize some special classes of Toeplitz codes as LCD codes (resp. linear codes with one-dimensional hull or self-orthogonal codes). It is worth mentioning that we determine the hull dimension of the isometric image of a linear code. Some optimal or almost optimal binary linear codes are obtained as a result. Most importantly, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) from quasi-cyclic codes. In particular, we construct binary LCD [32,16,8], [33,11,11] and [34,17,8] codes with improved parameters directly.