Dual Code Research Articles

Let R be a commutative local finite ring. In this paper, we construct the complete set of pairwise orthogonal primitive idempotents of R[X]/ <g> where g is a regular polynomial in R[X]. We use this set to decompose the ring R[X]/ <g> and to give the structure of constacyclic codes over finite chain rings. This allows us to describe generators of the dual code C' of a constacyclic code C and to characterize non-trivial self-dual constacyclic codes over finite chain rings.

Linear complementary dual (LCD) codes and linear complementary pair (LCP) of codes over finite fields have been intensively studied recently due to their applications in cryptography, in the context of side channel and fault injection attacks. The security parameter for an LCP of codes (C, D) is defined as the minimum of the minimum distances d(C) and $$d(D^\bot )$$ . It has been recently shown that if C and D are both 2-sided group codes over a finite field, then C and $$D^\bot $$ are permutation equivalent. Hence the security parameter for an LCP of 2-sided group codes (C, D) is simply d(C). We extend this result to 2-sided group codes over finite chain rings.

If we fix the code length n and dimension k, maximum distance separable (briefly, MDS) codes form an important class of codes because the class of MDS codes has the greatest error-correcting and detecting capabilities. In this paper, we establish all MDS constacyclic codes of length ps over $\mathbb {F}_{p^{m}}$ . We also give some examples of MDS constacyclic codes over finite fields. As an application, we construct all quantum MDS codes from repeated-root codes of prime power lengths over finite fields using the CSS and Hermitian constructions. We provide all quantum MDS codes constructed from dual codes of repeated-root codes of prime power lengths over finite fields using the Hermitian construction. They are new in the sense that their parameters are different from all the previous constructions. Moreover, some of them have larger Hamming distances than the well known quantum error-correcting codes in the literature.

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].

Abstract This essay constitutes a close reading of the works of Feofan Prokopovich that touch upon gender and womanhood. Interpretively it is informed by Judith Butler’s book Gender Trouble, specifically by her model of gender-as-performance. Prokopovich’s writings conveyed a negative characterization of holy women and Russian women of power, a combination of glaring silences and Scholastic dual codes that in toto denied the association of womanhood with glory or wisdom. In this he stood apart from other East Slavic Orthodox homilists of his day, even though they too invariably associated virtue with masculinity (muzhestvo). For Prokopovich, wisdom, strength, constancy, etc., were innately masculine. Women, by contrast, were weak, inconstant, non-rational, and guided by emotion. His sermons nominally in praise of Catherine I and Anna Ioannovna were suffused with narrative gestures that, to those attuned to the nuances of Scholastic rhetoric, ran entirely counter to their nominal message. Several panegyrics to Anna, for example, made no mention of her at all, a practice in sharp contrast to his sermons to male rulers, which typically placed the honoree firmly in the foreground. Even more startling is his singularly minimalist approach to Mary, for whom he composed almost no sermons and whose presence he barely mentioned in tracts where one would have expected otherwise. This essay concludes that this attitude reflected both his personal preferences and influence that Protestant Pietism had on his thinking.

Secret sharing has been a subject of study since 1979. In the secret sharing schemes there are some participants and a dealer. The dealer chooses a secret. The main principle is to distribute a secret amongst a group of participants. Each of whom is called a share of the secret. The secret can be retrieved by participants. Clearly the participants combine their shares to reach the secret. One of the secret sharing schemes is threshold secret sharing scheme. A threshold secret sharing scheme is a method of distribution of information among participants such that can recover the secret but cannot. The coding theory has been an important role in the constructing of the secret sharing schemes. Since the code of a symmetric design is a linear code, this study is about the multisecret-sharing schemes based on the dual code of code of a symmetric design. We construct a multisecret-sharing scheme Blakley's construction of secret sharing schemes using the binary codes of the symmetric design. Our scheme is a threshold secret sharing scheme. The access structure of the scheme has been described and shows its connection to the dual code. Furthermore, the number of minimal access elements has been formulated under certain conditions. We explain the security of this scheme.

Linear codes from vectorial Boolean power functions

Let $$\mathbb {F}_q$$ be a finite field with q elements, $$D_{2n,\,r}$$ a generalized dihedral group with $$\gcd (2n,q)=1$$ , and $$\mathbb {F}_q[D_{2n,\,r}]$$ a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of $$\mathbb {F}_q[D_{2n,\,r}]$$ is determined, which extends the results of Brochero Martinez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in $$\mathbb {F}_q[D_{2n,\,r}]$$ are precisely described and counted. Some numerical examples are also presented to illustrate our main results.

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.

We classify all the cyclic self-dual codes of length $$p^k$$ over the finite chain ring $$\mathcal R:=\mathbb Z_p[u]/\langle u^3 \rangle $$ , which is not a Galois ring, where p is a prime number and k is a positive integer. First, we find all the dual codes of cyclic codes over $${\mathcal R}$$ of length $$p^k$$ for every prime p. We then prove that if a cyclic code over $${\mathcal R}$$ of length $$p^k$$ is self-dual, then p should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ . Finally, we obtain a mass formula for counting cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ .

Abstract The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasisymmetric 2‐ and 2‐ designs with intersection numbers 0 and 3, and the nonexistence of a 2‐ quasi‐3 design. The nonexistence of a 2‐ quasi‐3 design is also proved.

This paper analyses for the first time the impact of new GPS signals on positioning accuracy for dynamic urban applications, taking bus operations as an example. The performance assessment addresses both code measurement precision and positioning accuracy. The former is based on signal-to-noise ratio and estimation of multipath and noise by a combination of code and carrier phase measurements. The impact on positioning accuracy is derived by comparing the performance achievable with the conventional single frequency GPS only positioning both relative to reference trajectories from the integration of carrier phase measurements with data from a high grade inertial measurement unit. The results show that L5 code measurements have the highest precision, followed by L1 C/A and L2C. In the positioning domain, there is a significant improvement in two-dimensional and three-dimensional accuracy from dual frequency code measurements over the single frequency measurements, of 39% and 48% respectively, enabling more bus operation services to be supported.

Due to some practical applications, linear complementary dual (LCD) codes and self-orthogonal codes have attracted wide attention in recent years. In this paper, we use simplicial complexes for construction of an infinite family of binary LCD codes and two infinite families of binary self-orthogonal codes. Moreover, we explicitly determine the weight distributions of these codes. We obtain binary LCD codes which have minimum weights two or three, and we also find some self-orthogonal codes meeting the Griesmer bound. As examples, we also present some (almost) {\it optimal} binary self-orthogonal codes and LCD {\it distance optimal} codes.

In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 ⟩ where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .

Remark on subcodes of linear complementary dual codes

For a Distributed Storage System (DSS), the Fractional Repetition (FR) code is a class in which replicas of encoded data packets are stored on distributed chunk servers, where the encoding is done using the Maximum Distance Separable (MDS) code. The FR codes allow for the exact uncoded repair with minimum repair bandwidth. In this paper, FR codes (called Flower codes) are constructed using finite binary sequences. It is shown that, for any FR code, there exists a Flower code and therefore Flower code is the general framework to construct FR code with uniform as well as non-uniform parameters. The condition for universally good Flower code is calculated on such sequences. For some sequences, the universally good Flower codes and Locally Repairable Flower codes are explored. In addition, conditions for equivalent Flower codes and dual Flower codes are also investigated in this paper. Some families of Flower codes with non-uniform parameters are obtained such that, from those families, Flowers code with uniform parameters are optimal FR codes in the literature. It is shown that any FR code is a Flower code and some known FR codes are obtained as the special cases of Flower codes using sequences.

ABSTRACT Linear complementary dual (LCD) codes are linear codes satisfying . Researchers have proved to construct linear codes over finite fields , q>3, is equivalent to construct LCD codes. This means that the investigation of binary and ternary LCD codes is the only remaining open problem. In this work, we propose new constructions of binary and ternary LCD codes by using matrix-product codes and prove that binary and ternary LCD codes from matrix-product codes are asymptotically good. Then we construct some new and good binary and ternary LCD codes using matrix-product matrices.

A note on linear complementary pairs of group codes

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

Output Planning at the Input Stage: Action Imprinting for Future Memory-Guided Behaviour