Double Circulant Codes Research Articles

Linear complementary dual codes and double circulant codes over a semi-local ring

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> be an odd prime power and <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_q $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M3">\begin{document}$ q $\end{document}</tex-math></inline-formula> elements. In this paper, suppose ring <inline-formula><tex-math id="M4">\begin{document}$ R = \mathbb{F}_{q}+ \mu \mathbb{F}_{q}+ \nu \mathbb{F}_{q}+ \mu \nu \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \mu \nu = \nu \mu, \mu^{2} = \mu, \nu^{2} = \nu. $\end{document}</tex-math></inline-formula> We first give a Gray map from <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula> onto <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_q^{4} $\end{document}</tex-math></inline-formula> and consider a decomposition of the ring <inline-formula><tex-math id="M8">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Additionally, we investigate linear complementary dual (LCD) codes over the ring <inline-formula><tex-math id="M9">\begin{document}$ R $\end{document}</tex-math></inline-formula>. Some conditions for such linear codes over <inline-formula><tex-math id="M10">\begin{document}$ R $\end{document}</tex-math></inline-formula> to be linear complementary dual are given. Furthermore, based on the Artin conjecture, we get a class of good codes by calculating the total number of LCD double circulant codes over <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula>.</p>

Double circulant codes for the Lee and Euclidean distance

<abstract><p>This paper investigates double circulant codes of length $ 2n $ over $ \mathbb{Z}_{{{p^m}}} $ where $ p $ is an odd prime, $ n $ goes to infinity, and $ m\ge 1 $ is a fixed integer. Using random coding, we obtain families of asymptotically good Lee codes over $ \mathbb{Z}_{{{p^m}}} $ in the case of small and large alphabets, and asymptotically good Euclidean codes over $ \mathbb{Z}_{{{p^m}}} $ for small alphabets. We use Euclidean codes to construct spherical codes, and Lee codes to construct insertion/deletion codes, by a projection technique due to (Yaglom, 1958) for spherical codes, and to (Sok et al., 2018) for deletion codes.</p></abstract>

Self-dual 2-quasi-cyclic codes and dihedral codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if charF=2), or a consta-dihedral code (if charF≠2). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other.

Self-dual and LCD double circulant codes over a class of non-local rings

Self-dual and LCD double circulant codes over a class of non-local rings

An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23)

This paper gives new methods of constructing symmetric self-dual codes over a finite field GF(q) where q is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using these methods, we construct an amount of symmetric self-dual codes over GF(11), GF(19), and GF(23) of every length less than 42. We also find 153 new self-dual codes up to equivalence: they are [32, 16, 12], [36, 18, 13], and [40, 20, 14] codes over GF(11), [36, 18, 14] and [40, 20, 15] codes over GF(19), and [32, 16, 12], [36, 18, 14], and [40, 20, 15] codes over GF(23). They all have new parameters with respect to self-dual codes. Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Let q be an odd prime power, and denote by $${\mathbb {F}}_q$$ the finite field with q elements. In this paper, we consider the ring $$R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$ , where $$u^2=u, v^2=v,uv=vu=0$$ and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over $${\mathbb {F}}_5$$ as the Gray images of these codes over R are given in short lengths.

Construction of isodual codes from polycirculant matrices

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over $$ {\mathbb {F}}_2$$ in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over $${\mathbb {F}}_2,$$ we show that binary double polycirculant codes are asymptotically good.

Generalized Cyclotomic Double Circulant Self-Dual Codes of Length 4p

Constructions of some double circulant self-dual codes on the residue ring $Z_{2p}$ , where $p$ is a prime and satisfies $p \equiv 3 \bmod {4}$ , are proposed. Two new structures of these codes are formed from the theory of generalized cyclotomy. Moreover, both [12, 6, 4] and [44, 22, 8] optimal self-dual codes over GF(2) and a family of [28, 14, 9] highest known self-dual codes over GF(4) are given.

On Self‐dual and LCD Double Circulant Codes over a Non‐chain Ring*

In this paper, we study double circulant codes of length 2n over the non-chain ring R = 𝔽 q + v𝔽 q + v 2𝔽 q ; where q is an odd prime power and v 3 = v. Exact enumerations of self-dual and LCD double circulant codes of length 2n over R are derived. When n is an odd prime, using random coding, we obtain families of asymptotically good self-dual and LCD codes of length 6n over 𝔽 q .

On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$

Double circulant codes of length 2n over the non-local ring $R=\mathbb {F}_{q}+u\mathbb {F}_{q}, u^{2}=u,$ are studied when q is an odd prime power, and − 1 is a square in $\mathbb {F}_{q}$. Double negacirculant codes of length 2n are studied over R when n is even, and q is an odd prime power. Exact enumeration of self-dual and LCD such codes for given length 2n is given. Employing a duality-preserving Gray map, self-dual and LCD codes of length 4n over $\mathbb {F}_{q}$ are constructed. Using random coding and the Artin conjecture, the relative distance of these codes is bounded below for n →∞. The parameters of examples of modest lengths are computed. Several such codes are optimal.

Double circulant self-dual and LCD codes over Galois rings

This paper investigates the existence, enumeration and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic $p^2$ and order $p^4$ with $p$ and odd prime. When $p \equiv 3 \pmod{4},$ we give an algorithm to construct a duality preserving bijective Gray map from such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Using random coding, we obtain families of asymptotically good self-dual and LCD codes over $\mathbb{Z}_{p^2},$ for the metric induced by the standard $\mathbb{F}_p$-valued Gray maps.

Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over $\F_q,$ for many classes of $q$'s. This is a partial solution to a fifty year old open problem.

A New Way to Find the Minimum Distance of BCH and Quadratic Double Circulant Codes

BCH (Bose, Chaudhuri, Hocquenghem) and QDC (Quadratic Double Circulant) codes are important classes of linear codes with many application domains. For large lengths, the true value of the minimum distances of these codes remains undetermined and therefore their error correcting capabilities are still unknown. The problem of searching codewords of lowest weight in linear codes is NP-hard problem and it is equivalent to the problem of the minimum distance research. Many methods have been used to attack this hardness such as genetic algorithms, simulated annealing, Multiple Impulse Method (MIM), Chen algorithm, Canteaut-Chabaud and Zimmermann algorithms. In a previous work, we have presented the method MIM-RSC to find the minimum weights by applying the MIM method on some Random sub codes of reduced dimensions. This reduction allows an efficient local search, but the important question here is which are the sub codes containing the lowest weights and how find them?. In general, the answer of this question isn’t evident and consequently many random sub codes should be exploited to have a great chance for taking a true minimum weight codeword. In this work, we present a new way to catch lowest weights by applying the Zimmermann algorithm on these random sub codes. This proposed scheme, Zimmermann-RSC, is validated on some BCH and QDC codes of known minimum weight and it is used to determine the minimum distance for many other codes of large lengths belonging to these families. The comparison between some competitors methods with the proposed scheme Zimmermann-RSC on many large BCH and QDC codes proves the efficiency of this latest in terms of the results quality and run time.

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them, generalizing some old results of MacWilliams on the enumeration of circulant orthogonal matrices. These formulae, in turn, are instrumental in deriving a Varshamov–Gilbert bound on the relative minimum distance of this family of codes.

On self-dual double negacirculant codes

Double negacirculant (DN) codes are the analogues in odd characteristic of double circulant codes. Self-dual DN codes are shown to have a transitive automorphism group. Exact counting formulae are derived for DN codes. The special class of length a power of two is studied by means of Dickson polynomials, and is shown to contain families of codes with relative distances satisfying a modified Varshamov–Gilbert bound. This gives an alternative, and effective proof of the result of Chepyzhov, that there are families of quasi-twisted codes above Varshamov–Gilbert.

5-SEEDs from the lifted Golay code of length 24 over Z<sub>4</sub>

Spontaneous emission error designs (SEEDs) are combinatorial objects that can be used to construct quantum jump codes. The lifted Golay code $G_{24}$ of length $24$ over $\mathbb{Z}_4$ is cyclic self-dual code. It is known that $G_{24}$ yields $5$-designs. In this paper, by using the generator matrices of bordered double circulant codes, we obtain $22$ mutually disjoint $5$-$(24, k, \lambda)$ designs with $(k, \lambda)=(8, 1), $ $(10, 36), $ $(12,1584)$ and $5$-$(24, k;22)$-SEEDs for $k=8, $ $10, $ $12, $ $13$ from $G_{24}$.

On some double circulant codes of Crnković and disproof of his two conjectures

Crnković (2014) introduced a self-orthogonal [2q,q−1] code and a self-dual [2q+2,q+1] code over the finite field Fp arising from orbit matrices for Menon designs, for every prime power q, where q≡1(mod4) and p a prime dividing q+12. He showed that if q is a prime and q=12m+5, where m is a non-negative integer, then the self-dual [2q+2,q+1] code over F3 is equivalent to a Pless symmetry code. However for other values of q, he remarked that these codes, up to his knowledge, do not belong to some previously known series of codes. In this paper, we describe an equivalence between his self-dual codes and the known codes introduced by Gaborit in 2002. On the other hand, Crnković (2014) also conjectured that if p=q+12 is a prime, the self-orthogonal code and the self-dual code have minimum distance p+3. We disprove this conjecture by giving two counter-examples in the case of the self-orthogonal code and the self-dual code, respectively when q=25 and p=13.

Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes

It is known that extremal doubly-even self-dual codes of length $$n\equiv 8$$ n ? 8 or $$0\ (\mathrm {mod}\ 24)$$ 0 ( mod 24 ) yield 3- or 5-designs respectively. In this paper, by using the generator matrices of bordered double circulant doubly-even self-dual codes, we give 3-(n, k; m)-SEEDs with (n, k, m) $$\in \{(32,8,5), (56,12,9), (56,16,9), (56,24,9), (80,16,52)\}$$ ? { ( 32 , 8 , 5 ) , ( 56 , 12 , 9 ) , ( 56 , 16 , 9 ) , ( 56 , 24 , 9 ) , ( 80 , 16 , 52 ) } . With the aid of computer, we obtain 22 generator matrices of bordered double circulant doubly-even self-dual codes of length 48, which enable us to get 506 mutually disjoint 5-(48, k, $$\lambda $$ ? ) designs for (k, $$\lambda $$ ? )=(12, 8),(16, 1356),(20, 36176). Moreover, this implies 5-(48, k; 506)-SEEDs for $$k=12, 16, 20, 24$$ k = 12 , 16 , 20 , 24 .

A New Class of Codes for Boolean Masking of Cryptographic Computations

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length $<132.$ We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths $\le 12$ by the building up construction. Some nonlinear permutations are constructed by using $\Z_4$-codes, based on the notion of dual distance of an unrestricted code.

Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over \(\mathbb{F}_{2}\), \(\mathbb{F}_{3}\), \(\mathbb{F}_{4}\), \(\mathbb{F}_{5}\), and \(\mathbb{F}_{7}\). In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.