Dual Code Research Articles

Negacyclic codes of prime power length over the finite non-commutative chain ring 𝔽pm[u,𝜃] 〈u2〉

In this paper, we focus on the algebraic structure of left negacyclic codes of length [Formula: see text] over the finite non-commutative chain ring [Formula: see text] where [Formula: see text] is an automorphism on [Formula: see text]. After that, the number of codewords of all left negacyclic codes is obtained. For each left negacyclic code, we also obtain the structure of its right dual code. In the remaining result, the number of distinct left negacyclic codes is given. Finally, a one-to-one correspondence between left negacyclic and left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] is constructed via ring isomorphism, which carries over the results regarding left negacyclic codes corresponding to left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] is a nonzero element of the field [Formula: see text] such that [Formula: see text].

Cyclic codes of length pn over (ℤ p ) m

The purpose of this paper is describing cyclic codes of length pn over (ℤ p ) m . We partition ideals of into two general forms and calculate the annihilator of these two forms. In this way, we can introduce an explanation of cyclic codes and the dual code of an arbitrary cyclic code of length pn over (ℤ p ) m .

Several constructions of LCD MDS codes

Linear complementary dual codes (LCD codes) have been widely studied because of their significant role in data storage and cryptography. Maximum distance separable codes with complimentary duals (LCD MDS codes) have the greatest error correcting ability, and so the construction of LCD MDS codes is a hot topic. By using the generator matrix of linear codes, the existence results of LCD MDS codes are given by Carlet et al. in 2018. Recently, several display constructions of LCD MDS codes are given by scholars via the generalized Reed-Solomoncodes, the algebraic geometry codes, the constacylic codes and so on. In this paper, we will introduce the main methods, results and recent progress of constructing LCD MDS codes.

New quantum codes from cyclic codes over finite chain ring of length 3

Let [Formula: see text] be a commutative finite chain ring with [Formula: see text] is its maximal ideal, [Formula: see text] is its residual field and [Formula: see text] the index of nilpotency of [Formula: see text]. In this paper, we compute the dual code of some types of cyclic codes over [Formula: see text] when [Formula: see text]. Afterward, we show how extract some new quantum codes from these cyclic codes.

On [formula omitted]-cyclic codes and their applications in constructing optimal codes

Let R=F2+uF2(u2=0) and S=F2+uF2+u2F2(u3=0) be two finite commutative chain rings. This paper studies F2RS-cyclic codes, which are described as S[x]-submodules of the S[x]-module F2[x]∕〈xr−1〉×R[x]∕〈xs−1〉×S[x]∕〈xt−1〉. We study their generator polynomials and the minimal generating sets. We classify each case of the generating sets separately and determine the size of each such case. Free F2RS-cyclic codes and separable codes are discussed, and the structural properties of dual codes of free F2RS-cyclic codes are investigated. Moreover, we determine the relationship between the generator polynomials of free F2RS-cyclic codes and their duals. As applications, we provide several examples of optimal and near-optimal binary codes which are obtained from the Gray images of F2RS-cyclic codes.

Quantum Codes and Entanglement-Assisted Quantum Codes Derived from One-Generator Quasi-Twisted Codes

In this paper, we consider a special class of one-generator quasi-twisted (QT) codes of index 2 and explore their algebraic structure. By studying the form of Hermitian dual codes, a sufficient condition for self-orthogonality with Hermitian inner product is proposed. Using some one-generator QT codes, we construct some quantum codes on small finite fields. Moreover, a new method is provided to construct maximal-entanglement entanglement-assisted quantum codes via the class of QT codes. By comparison, our quantum codes and maximal-entanglement entanglement-assisted quantum codes have better parameters than any known codes.

Cyclic and constacyclic codes over the ring Z4[u]/⟨u3 − u2⟩ and their Gray images

In this article, the structure of generator polynomial of the cyclic codes with odd length is formed over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ where $u^3=u^2$. With the isomorphism we have defined, the generator polynomial of constacyclic codes with odd length over this ring is created from the generator of the cyclic codes. Additionally, necessary and sufficient conditions for a linear code in this ring to be a self dual code and a LCD code are mentioned. Furthermore, for all units over this ring, $\mathbb{Z}_{4}$-images of $\lambda$-constacyclic codes and also $\mathbb{Z}_{4}$-images of cyclic codes are examined by using related ones from defined three new Gray maps. Moreover, several new and optimal codes are constructed in terms of the Lee, Euclidean and Hamming weight in reference to the database.

Characterization and classification of optimal LCD codes

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over $$\mathbb {F}_q$$ having large minimum weights for $$q \in \{2,3\}$$ . Using the characterization, for arbitrary n, we determine the largest minimum weights among LCD [n, k] codes over $$\mathbb {F}_q$$ , where $$(q,k) \in \{(2,4), (3,2),(3,3)\}$$ . Moreover, for arbitrary n, we give a complete classification of optimal LCD [n, k] codes over $$\mathbb {F}_q$$ , where $$(q,k) \in \{(2,3), (2,4), (3,2),(3,3)\}$$ .

Small weight codewords of projective geometric codes

We investigate small weight codewords of the p-ary linear code Cj,k(n,q) generated by the incidence matrix of k-spaces and j-spaces of PG(n,q) and its dual, with q a prime power and 0⩽j<k<n. Firstly, we prove that all codewords of Cj,k(n,q) up to weight (3−O(1q))[k+1j+1]q are linear combinations of at most two k-spaces (i.e. two rows of the incidence matrix). As for the dual code Cj,k(n,q)⊥, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight codewords (2) to the case C0,1(n,q)⊥. This implies the solution to both problem (1) and (2) if q is prime and the solution to problem (1) if q is even.

(σ, δ)-Skew quasi-cyclic codes over the ring $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$

Let $R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ be a finite non-chain ring, where u2 = 1. In this paper, we consider (σ, δ)-skew quasi-cyclic codes over the ring R, where σ is an automorphism of R and δ is an inner σ-derivation of R. We determine the structure of 1-generator (σ, δ)-skew quasi-cyclic codes over R and give a sufficient condition for 1-generator (σ, δ)-skew quasi-cyclic codes over R to be free. We also determine a distance bound for free 1-generator (σ, δ)-skew quasi-cyclic codes. Moreover, using the residue codes of these codes over R we obtain some good $\mathbb {Z}_{4}$ -linear codes. Finally, we give the characterization of Euclidean dual codes of (σ, δ)-skew quasi-cyclic codes.

On the weights of dual codes arising from the GK curve

In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti–Korchmáros maximal curve. We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized Hamming weights of such codes.

Hermitian Rank Metric Codes and Duality

In this paper we define and study rank metric codes endowed with a Hermitian form. We analyze the duality for $\mathbb {F}_{q^{2}}$ -linear matrix codes in the ambient space $(\mathbb {F}_{q^{2}})_{n,m}$ and for both $\mathbb {F}_{q^{2}}$ -additive codes and $\mathbb {F}_{q^{2m}}$ -linear codes in the ambient space $\mathbb {F}_{q^{2m}}^{n}$ . Similarly, as in the Euclidean case we establish a relationship between the duality of these families of codes. For this we introduce the concept of $q^{m}$ -duality between bases of $\mathbb {F}_{q^{2m}}$ over $\mathbb {F}_{q^{2}}$ and prove that a $q^{m}$ -self dual basis exists if and only if $m$ is an odd integer. We obtain connections on the dual codes in $\mathbb {F}_{q^{2m}}^{n}$ and $(\mathbb {F}_{q^{2}})_{n,m}$ with the corresponding inner products. In particular, we study Hermitian linear complementary dual, Hermitian self-dual and Hermitian self-orthogonal codes in $\mathbb {F}_{q^{2m}}^{n}$ and $(\mathbb {F}_{q^{2}})_{n,m}$ . Furthermore, we present connections between Hermitian $\mathbb {F}_{q^{2}}$ -additive codes and Euclidean $\mathbb {F}_{q^{2}}$ -additive codes in $\mathbb {F}_{q^{2m}}^{n}$ .

On the algebraic structure of polycyclic codes

In this paper, we are interested in the study of the right polycyclic codes as invariant subspaces of Fnq by a fixed operator TR. This approach has helped in one hand to connect them to the ideals of the polynomials ring Fq [x]/?f)X)?, where f (x) is the minimal polynomial of TR. On the other hand, it allows to prove that the dual of a right polycyclic code is invariant by the adjoint operator of TR. Hence, when TR is normal we prove that the dual code of a right polycyclic code is also a right polycyclic code. However, when TR isn?t normal the dual code is equivalent to a right polycyclic code. Finally, as in the cyclic case, the BCH-like and Hartmann-Tzeng-like bounds for the right polycyclic codes on Hamming distance are derived.

Weight distributions for projective binary linear codes from Weil sums

<abstract> A class of projective binary linear codes are constructed and their weight distributions are investigated using Weil sums. They have at most three nonzero weights, containing some optimal codes. Their dual codes are also studied and some of them are either optimal or almost optimal. </abstract>

On Z₂Z₂[u³]-Additive Cyclic and Complementary Dual Codes

Aydogdu et al. studied the standard forms of generator and parity-check matrices of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, and presented generators of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes (Finite Fields Appl. 48: 241-260, 2017). In this paper, we investigate some other useful properties of $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive codes, including asymptotically good $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2}[u^{3}]$ -additive complementary dual codes. The present paper can be viewed as a necessary complementary part of Aydogdu’s work.

Memory Transfer Language as a Tool for Visualization-Based-Pedagogy

In this paper MTL, as a tool for visualization-based pedagogy, is discussed. MTL approach in teaching and learning programming is based on Dual Code Theory (DCT) whereby lectures, tutorials, laboratory sessions and individual study are all carried out using two systems (verbal and non-verbal). For visualization (non-verbal) one single visualization scheme (RAM) models together with animations) is used to represent the images while text and voice narratives are used for the verbal presentation of lessons. A class experiment is carried out to evaluate the impact of using MTL strongly and longitudinally where lectures, tutorials, labs, group discussions and assignments are perfumed using RAM models and celiot animation. The results of the chi-square test revealed that students’ performance on one (question 1) was significantly (p<0.0001) higher (experimental group 23.53% as compared to control 1.89%). Similarly, the results of the chi-square test presented revealed that students’ performance on another (question 2) was significantly (p<0.0001) higher (experimental group 23.53% as compared to control 1.14%). For much better results, changing students’ learning techniques is recommended to precede strong and longitudinal use of MTL in teaching and learning programming.

Decoding Algorithm by Cooperation Between Hartmann Rudolph Algorithm and a Decoder Based on Syndrome and Hash

In this paper, the authors present a concatenation of Hartmann and Rudolph (HR) partially exploited and a decoder based on hash techniques and syndrome calculation to decode linear block codes. This work consists firstly to use the HR with a reduced number of codewords of the dual code then the HWDec which exploits the output of the HR partially exploited. Researchers have applied the proposed decoder to decode some Bose, Chaudhuri, and Hocquenghem (BCH) and quadratic residue (QR) codes. The simulation and comparison results show that the proposed decoder guarantees very good performances, compared to several competitors, with a much-reduced number of codewords of the dual code. For example, for the BCH(31, 16, 7) code, the good results found are based only on 3.66% of the all codewords of the dual code space, for the same code the reduction rate of the run time varies between 78% and 90% comparing to the use of Hartmann and Rudolph alone. This shows the efficiency, the rapidity, and the reduction of the memory space necessary for the proposed concatenation.

An explicit construction of quantum codes from one-generator generalized quasi-cyclic codes

In this paper, we take advantage of a class of one-generator generalized quasi-cyclic (GQC) codes of index 2 to construct quantum error-correcting codes. By studying the form of Hermitian dual codes and their algebraic structure, we propose a sufficient condition for self-orthogonality of GQC codes with Hermitian inner product. By comparison, the quantum codes we constructed have better parameters than known codes.

On the equivalence of authentication codes and robust (2, 2)-threshold schemes

Abstract In this paper, we show a “direct” equivalence between certain authentication codes and robust threshold schemes. It was previously known that authentication codes and robust threshold schemes are closely related to similar types of designs, but direct equivalences had not been considered in the literature. Our new equivalences motivate the consideration of a certain “key-substitution attack.” We study this attack and analyze it in the setting of “dual authentication codes.” We also show how this viewpoint provides a nice way to prove properties and generalizations of some known constructions.

Cyclic LRC-LCD Codes With Hierarchical Locality

Locally recoverable (LRC) codes and linear complementary dual (LCD) codes are both useful in distributed storage systems. In this letter, we establish a connection between LRC codes and LCD codes. We derive some conditions on the construction of cyclic codes with $(r,\delta)$ locality and codes with hierarchical locality so that they are also LCD codes. Some constructed codes are optimal. These codes may be useful in simultaneously providing local recoverability and security against certain attacks in distributed storage systems.