Narrow Results

Let ${𝔽}_q$ be an arbitrary finite field and $n\ge 2$ be an integer with $\mathrm{\text{gcd}}(n,q)=1$. We study all${𝔽}_q$-linear additive cyclic codes over ${𝔽}_{q^3}$ of length $n$systematically. This is much more complicated compared to the same task over ${𝔽}_{q^2}$. We obtain a canonical unique representation. We explicitly obtain the dual codes in the canonical form under the Euclidean and trace the Galois inner products. We characterize and construct large classes of complementary dual codes among ${𝔽}_q$-linear additive cyclic codes over ${𝔽}_{q^3}$ of length $n$ under the trace Euclidean and the trace Galois inner products. We obtain interesting differences depending on the canonical representation and also on the inner products. We also study subfield subcodes and trace (onto ${𝔽}_q$) codes.

In this paper, we study linear codes of length $n=m\ell$ that are invariant under an endomorphism $T=T_1\oplus T_1\oplus \cdots \oplus T_1$ ($\ell$ copies of $T_1$), where $T_1$ is a cyclic endomorphism on ${𝔽}_q^m$. As each endomorphism can be represented by a matrix, we restrict our study on linear codes that are under a matrix $D=\mathrm{\text{diag}}(M,M,\dots ,M)$, where $M$ is an $m\times m$ cyclic matrix, called quasi-$M$-cyclic codes of index $\ell$, and quasi-$f$-cyclic codes when $M$ is the companion matrix of a polynomial $f(x)$. We prove a one-to-one correspondence between quasi-$M$-cyclic codes of index $\ell$ and $R_f$-submodules of $R_f^{\ell }$, where $R_f={𝔽}_q[x]/〈f(x)〉$ and $f(x)$ is the minimal polynomial of $M$. We prove the BCH-like and Hartmann–Tzeng-like bounds for $1$-generator quasi-$f$-cyclic codes. In addition, we study the additive structure of quasi-$f$-cyclic codes by mapping them to ${𝔽}_{q^{\ell }}/〈f(x)〉$ via an ${𝔽}_{q^{\ell }}$-module morphism. Finally, we provide examples of new quantum codes derived from quasi-$f$-cyclic codes as an application of our results.

A verifiable multi-secret sharing (VMSS) scheme addresses various security issues, helping in cheating detection and cheater identification. The dimension of the intersection of a given pair of cyclic codes over a finite field ${𝔽}_q$ is represented by $\ell$. In this paper, we propose a VMSS scheme based on an $\ell$-intersection pair of cyclic codes. The proposed scheme is very useful for sharing a large number of secrets. If $p$ is the number of secrets in a threshold $(t,n)$ scheme, then for $p>t$, the computational complexity and the public values required in the reconstruction phase are much less compared to many existing schemes in the literature. Overall, the proposed scheme comprises nearly all the features, like being verifiable, testing participants’ integrity, checking Verifier frauds, and most importantly, it is of multi-use type.

Using Gröbner techniques, we can exhibit a method to get the distance and weight distribution of cyclic codes and shortened cyclic codes, improving earlier similar results for the distance of cyclic codes.

We give the structure of λ-constacyclic codes of length p^sm over R = 𝔽_p^r[u]/〈u^e〉 with . We also give the structure of λ-constacyclic codes of length p^sm with λ = α₁ + uα₂ + ⋯ + u^e-1 α_e-1, and study the self-duality of these codes.

The explicit expressions of generator polynomials of cyclic codes of length $2^n$ over finite fields are obtained. The coefficients of these generator polynomials and check polynomials are obtained through modular Lucas sequences. Further, using these polynomials, self-dual, reversible and self-orthogonal cyclic codes of length $2^n$ are classified.

Let ${𝔽}_{p^m}$ be the finite field of order $p^m,$ where $p$ is an odd prime and $m$ is a positive integer. In this paper, we determine the algebraic structures of all cyclic and negacyclic codes of length $4p^s$ over the finite commutative chain ring ${ℛ}={𝔽}_{p^m}+u{𝔽}_{p^m},$ where $u^2=0$ and $s$ is a positive integer. We also obtain the number of codewords in each of these codes. Among others, we establish the duals of all such codes and derive some self-dual cyclic and negacyclic codes of length $4p^s$ over ${ℛ}.$

Let $p$ be an odd prime, and $k$ be an integer such that $gcd(k,p)=1$. Using pairwise orthogonal idempotents ${\gamma }_1,{\gamma }_2,{\gamma }_3$ of the ring ${ℛ}={𝔽}_p[u]/〈u^{k+1}-u〉$, with ${\gamma }_1+{\gamma }_2+{\gamma }_3=1$, ${ℛ}$ is decomposed as ${ℛ}={\gamma }_1{ℛ}\oplus {\gamma }_2{ℛ}\oplus {\gamma }_3{ℛ}$, which contains the ring $R={\gamma }_1{𝔽}_p\oplus {\gamma }_2{𝔽}_p\oplus {\gamma }_3{𝔽}_p$ as a subring. It is shown that, for ${\lambda }_0,{\lambda }_k\in {𝔽}_p$, ${\lambda }_0+u^k{\lambda }_k\in R$, and it is invertible if and only if ${\lambda }_0$ and ${\lambda }_0+{\lambda }_k$ are units of ${𝔽}_p$. In such cases, we study $({\lambda }_0+u^k{\lambda }_k)$-constacyclic codes over $R$. We present a direct sum decomposition of $({\lambda }_0+u^k{\lambda }_k)$-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a $({\lambda }_0+u^k{\lambda }_k)$-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over ${𝔽}_p$, with better parameters than existing ones, are constructed from cyclic and negacyclic codes over $R$.

In this paper, we study linear codes invariant under a cyclic endomorphism $T,$ called $T$-cyclic codes. Since every cyclic endomorphism can be represented by a cyclic matrix with respect to a given basis, all of these matrices are similar. For simplicity we restrict ourselves to linear codes invariant under the right multiplication by a cyclic matrix $M,$ that we call $M$-cyclic codes, and when $M$ is the companion matrix $C_f$ of a given nonzero polynomial $f(x)$ we call them $f$-cyclic codes. The similarity relation between matrices helps us find connections between $M$-cyclic codes and $f$-cyclic codes, where $f(x)$ is the minimal polynomial of $M.$ The class of $M$-cyclic codes contains cyclic codes and their various generalizations such as constacyclic codes, right and left polycyclic codes, monomial codes, and others. As common in the study of cyclic codes and their generalizations, we make use of the one-to-one correspondence between $M$-cyclic codes and ideals of the polynomial ring $R_{{}_f}:={𝔽}_{{}_q}[x]/〈f(x)〉,$ where $f(x)$ is the minimal polynomial of $M.$ This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials among others. Next, we study the duality of these codes, where we show that the $b$-dual of an $M$-cyclic code is an $M^{{}^{\ast }}$-cyclic code, where $M^{{}^{\ast }}$ is the adjoint matrix of $M$ with respect to $b,$ and we explore some important results on the duality of these codes. Finally, we give examples as applications of some of the results and we construct some optimal codes.

We construct new families of duadic codes and derive from them degenerate quantum codes. This provides a solution to a question raised by Aly, Klappenecker and Sarvepalli on the existence of quantum codes from duadic codes when the multiplicative order of q modulo n is even. We also characterize the affine-invariant maximal extended cyclic codes. Then by the CSS construction, we give a family of quantum code.

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. They generalize all major quantum error protection schemes, and therefore are especially versatile. This paper introduces numerous constructions of subsystem codes. It is shown how one can derive subsystem codes from classical cyclic codes. Methods to trade the dimensions of subsystem and co-subsystem are introduced that maintain or improve the minimum distance. As a consequence, many optimal subsystem codes are obtained. Furthermore, it is shown how given subsystem codes can be extended, shortened, or combined to yield new subsystem codes. These subsystem code constructions are used to derive tables of upper and lower bounds on the subsystem code parameters.

We give a construction for quantum codes from linear and cyclic codes over . We derive Hermitian self-orthogonal codes over as Gray images of linear and cyclic codes over . In particular, we use two binary codes associated with a cyclic code over of odd length to determine the parameters of the corresponding quantum code.

Let R = F₃ + vF₃ be a finite commutative ring, where v² = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F₃ as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

In this paper, quantum codes from cyclic codes over A₂ = F₂ + uF₂ + vF₂ + uvF₂, u² = u, v² = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A₂, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A₂ that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A₂.

We give a construction of quantum codes over ${𝔽}_q$ from cyclic codes over a finite non-chain ring ${𝔽}_q+v{𝔽}_q+v^2{𝔽}_q+v^3{𝔽}_q$, where $q=p^r$, p is a prime, $3|(p-1)$ and $v^4=v$.

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring $B_r=F_2+v_1{F}_2+\cdots +v_r{F}_2$, $v_i^2=v_i,v_i{v}_j=v_j{v}_i=0,1\le i,j\le r$ for $r\ge 1$. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over $B_r$ are obtained. A necessary and sufficient condition for cyclic code over $B_r$ that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over $B_r$.

Quantum key distribution (QKD) is a cryptographic communication protocol that utilizes quantum mechanical properties for provable absolute security against an eavesdropper. The communication is carried between two terminals using random photon polarization states represented through quantum states. Both these terminals are interconnected through disjoint quantum and classical channels. Information reconciliation using delay controlled joint decoding is performed at the receiving terminal. Its performance is characterized using data and error rates. Achieving low error rates is particularly challenging for schemes based on error correcting codes with short code lengths. This article addresses the decoding process using ordered statistics decoding for information reconciliation of both short and medium length Bose–Chaudhuri–Hocquenghem codes over a QKD link. The link’s quantum channel is modeled as a binary symmetric quantum depolarization channel, whereas the classical channel is configured with additive white Gaussian noise. Our results demonstrate the achievement of low bit error rates, and reduced decoding complexity when compared to other capacity achieving codes of similar length and configuration.

In this paper, we discuss the DNA construction of general length over the finite ring ${ℛ}={\mathbb{Z}}_4+v{\mathbb{Z}}_4$, with $v^2=v$, which plays a very significant role in DNA computing. We discuss the GC weight of DNA codes over ${ℛ}$. Several examples of reversible cyclic codes over ${ℛ}$ are provided, whose ${\mathbb{Z}}_4$-images are ${\mathbb{Z}}_4$-linear codes with good parameters.

For any odd prime p, a classification of all constacyclic codes of length 4p^s over 𝔽_p^m is obtained, which establishes the algebraic structure in term of specified polynomial generators of such codes. Among other results, all self-dual and LCD cyclic and negacylic codes of length 4p^s are obtained. As an example, all constacyclic codes of length 36 over 𝔽₂₇ and 𝔽₈₁ are listed.

Quasi-cyclic (QC) codes are a natural generalization of cyclic codes. In this paper, we study some structural properties of QC codes over $R={𝔽}_p+u{𝔽}_p$, where $p$ is a prime and $u^2=u$. By exploring their structure, we determine the one generator QC codes over $R$ and the size by giving a minimal spanning set. We discuss some examples of QC codes of various length over $R$.