Narrow Results

Let $p$ be a prime such that $p^m\equiv 1(mod4)$, where $m$ is a positive integer. For any nonzero element $\alpha$ of ${𝔽}_{p^m}$, we determine the algebraic structure of all $\alpha$-constacyclic codes of length $4p^s$ over the finite commutative chain ring $\frac{{𝔽}_{p^m}[u]}{〈u^3〉}\cong {𝔽}_{p^m}+u{𝔽}_{p^m}+u^2{𝔽}_{p^m}$, where $u^3=0$ and $s$ is a positive integer. If the unit $\alpha \in {𝔽}_{p^m}$ is a square, $\alpha ={\delta }^2$, each $\alpha$-constacyclic code of length $4p^s$ is expressed as a direct sum of an $-\delta$-constacyclic code and an $\delta$- constacyclic code of length $2p^s$. In the main case that the unit $\alpha$ is not a square, it is shown that any nonzero polynomial of degree at most $3$ over ${𝔽}_{p^m}$ is invertible in the ambient ring $\frac{({𝔽}_{p^m}+u{𝔽}_{p^m}+u^2{𝔽}_{p^m})[x]}{〈x^{4p^s}-\alpha 〉}$. It is also proven that the ambient ring $\frac{({𝔽}_{p^m}+u{𝔽}_{p^m}+u^2{𝔽}_{p^m})[x]}{〈x^{4p^s}-\alpha 〉}$ is a local ring with the unique maximal ideal $〈x^4-{\alpha }_0,u〉$, where ${\alpha }_0^{p^s}=\alpha$. Such $\alpha$-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each $\alpha$-constacyclic code are obtained. The non-existence of self-dual and isodual $\alpha$-constacyclic codes of length $4p^s$ over ${𝔽}_{p^m}+u{𝔽}_{p^m}+u^2{𝔽}_{p^m}$, when the unit $\alpha$ is not a square, is likewise proved.

Let $\lambda ={\lambda }_0+u{\lambda }_1+\cdots +u^{k-1}{\lambda }_{k-1}$ be a Type 1 unit in ${\Re }_k={𝔽}_{p^m}+u{𝔽}_{p^m}+\cdots +u^{k-1}{𝔽}_{p^m}(u^k=0),$ where $p$ is an odd prime, $m$ is a positive integer and ${\lambda }_0,{\lambda }_1,\dots ,{\lambda }_{k-1}\in {𝔽}_{p^m},{\lambda }_0\ne 0,{\lambda }_1\ne 0.$ In this paper, we give the complete structure of all Type 1 $\lambda$-constacyclic codes and their duals of length $8p^s$ over the finite commutative chain ring ${\Re }_k$ in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 $\lambda$-constacyclic codes. For $p^m\equiv 1(\mathrm{\text{mod}}4)$ and a unit $\lambda \in {\Re }_2,$ we determine the $b$ -symbol distances of all $\lambda$-constacyclic codes of length $8p^s$ over ${\Re }_2,$ where $b\le 8.$ As illustrations, we provide several $\lambda$-constacyclic codes with new parameters with respect to Hamming, RT and $b$-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS $b$-symbol codes are generalization of MDS codes. We found some MDS $b$-symbol constacyclic codes of length $8p^s$ over ${\Re }_2.$ Additionally, for $p^m\equiv 1(\text{mod}4),$ we provide a decoding algorithm for Type 1 constacyclic codes of length $8p^s$ over ${\Re }_k$ with respect to the Hamming, RT and $b$-symbol metrics.

In this paper, three families of quantum convolutional codes are constructed. The first one and the second one can be regarded as a generalization of Theorems 3, 4, 7 and 8 [J. Chen, J. Li, F. Yang and Y. Huang, Int. J. Theor. Phys., doi:10.1007/s10773-014-2214-6 (2014)], in the sense that we drop the constraint q ≡ 1 (mod 4). Furthermore, the second one and the third one attain the quantum generalized Singleton bound.

Let $p$ be a prime such that $p^m\equiv 3(mod4)$. For any unit $\lambda$ of ${𝔽}_{p^m}$, we determine the algebraic structures of $\lambda$-constacyclic codes of length $4p^s$ over the finite commutative chain ring ${𝔽}_{p^m}+u{𝔽}_{p^m}$, $u^2=0$. If the unit $\lambda \in {𝔽}_{p^m}$ is a square, each $\lambda$-constacyclic code of length $4p^s$ is expressed as a direct sum of an -$\alpha$-constacyclic code and an $\alpha$-constacyclic code of length $2p^s.$ If the unit $\lambda$ is not a square, then $x^4-{\lambda }_0$ can be decomposed into a product of two irreducible coprime quadratic polynomials which are $x^2+\gamma x+\frac{{\gamma }^2}{2}$ and $x^2-\gamma x+\frac{{\gamma }^2}{2}$, where ${\lambda }_0^{p^s}=\lambda$ and ${\gamma }^4=-4{\lambda }_0$. By showing that the quotient rings $\frac{{ℛ}}{〈\left(\right.x^2+\gamma x+\frac{{\gamma }^2}{2}{\left)\right.}^{p^s}〉}$ and $\frac{{ℛ}}{〈\left(\right.x^2-\gamma x+\frac{{\gamma }^2}{2}{\left)\right.}^{p^s}〉}$ are local, non-chain rings, we can compute the number of codewords in each of $\lambda$-constacyclic codes. Moreover, the duals of such codes are also given.

For any odd prime $p$ such that $p^m\equiv 3(mod4)$, the structures of all $(\alpha +u\beta )$-constacyclic codes of length $4p^s$ over the finite commutative chain ring ${𝔽}_{p^m}+u{𝔽}_{p^m}$$(u^2=0)$ are established in term of their generator polynomials. When the unit $(\alpha +u\beta )$ is a square, each $(\alpha +u\beta )$-constacyclic code of length $4p^s$ is expressed as a direct sum of two constacyclic codes of length $2p^s$. In the main case that the unit $(\alpha +u\beta )$ is not a square, it is shown that the ambient ring $\frac{({𝔽}_{p^m}+u{𝔽}_{p^m})[x]}{〈x^{4p^s}-(\alpha +u\beta )〉}$ is a principal ideal ring. From that, the structure, number of codewords, duals of all such $(\alpha +u\beta )$-constacyclic codes are obtained. As an application, we identify all self-orthogonal, dual-containing, and the unique self-dual $(\alpha +u\beta )$-constacyclic codes of length $4p^s$ over ${𝔽}_{p^m}+u{𝔽}_{p^m}$.

Let ${𝔽}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is an odd prime, $k,\lambda$ be positive integers satisfying $\lambda \ge 2$, and denote ${𝒦}={𝔽}_{p^m}[x]/〈f{(x)}^{\lambda p^k}〉$, where $f(x)$ is an irreducible polynomial in ${𝔽}_{p^m}[x]$. In this note, for any fixed invertible element $\omega \in {{𝒦}}^{\times }$, we present all distinct linear codes $S$ over ${𝒦}$ of length $2$ satisfying the condition: $(\omega f{(x)}^{p^k}{a}_1,a_0)\in S$ for all $(a_0,a_1)\in S$. This conclusion can be used to determine the structure of $(\delta +\alpha u^2)$-constacyclic codes over the finite chain ring ${𝔽}_{p^m}[u]/〈u^{2\lambda }〉$ of length $n{p}^k$ for any positive integer $n$ satisfying $\mathrm{\text{gcd}}(p,n)=1$.

Let $p$ be an odd prime, $s$ and $m$ be positive integers and $\lambda$ be a nonzero element of ${𝔽}_{p^m}$. The $\lambda$-constacyclic codes of length $p^s$ over ${𝔽}_{p^m}$ are linearly ordered under set theoretic inclusion as ideals of the chain ring ${𝔽}_{p^m}[x]/〈x^{p^s}-\lambda 〉$. Using this structure, the symbol-triple distances of all such $\lambda$-constacyclic codes are established in this paper. All maximum distance separable symbol-triple constacyclic codes of length $p^s$ are also determined as an application.

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 $R$.

Constacyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. Six new families of asymmetric quantum codes derived from constacyclic codes are constructed in this paper. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.

In this paper, we construct two classes of new quantum maximum-distance-separable (MDS) codes with parameters , where q is an odd prime power with q ≡ 3 (mod 4) and ; [[8(q - 1), 8(q - 1) - 2d + 2, d]]_q, where q is an odd prime power with the form q = 8t - 1 (t is an even positive integer) and . Comparing the parameters with all known quantum MDS codes, the quantum MDS codes exhibited here have minimum distances bigger than the ones available in the literature.

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of $q$-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths $n|(q^2-1)$ with flexible parameters as to the minimum distance $d$ and the number $c$ of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.

In this paper, we study λ-constacyclic codes over the ring R = ℤ₄ + uℤ₄, where u² = 0, for λ =1 + 3u and 3 + u. We introduce two new Gray maps from R to ${\mathbb{Z}}_4^4$ and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over ℤ₄. Moreover, we present many examples of λ-constacyclic codes over R whose ℤ₄-images have better parameters than the currently best-known linear codes over ℤ₄.

For any odd prime p such that p^m ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring ${ℛ}_a=\frac{{\mathbb{F}}_{p}m[u]}{〈u^a〉}={\mathbb{F}}_{p}m+u{\mathbb{F}}_{p}m+\cdots +u^{a-1}{\mathbb{F}}_{p}m$ that have the form Λ = Λ₀ + uΛ₁ + ⋯ + u^a−1 Λ_a−1, where Λ₀, Λ₁, …, Λ_a−1 ∊ 𝔽_pm, Λ₀ ≠ 0, Λ₁ ≠ 0. The class of Λ-constacyclic codes of length 4p^s over ℛ_a is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4p^s is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2p^s. In the main case that the unit Λ is not a square, we prove that the polynomial x⁴ − λ₀ can be decomposed as a product of two quadratic irreducible and monic coprime factors, where ${\text{λ}}_0^{p^s}={\Lambda }_0$. From this, the ambient ring $\frac{{ℛ}_a[x]}{〈x^{4p^s}-\Lambda 〉}$ is proven to be a principal ideal ring, whose maximal ideals are ⟨x² + 2ηx + 2η²⟩ and ⟨x² − 2ηx + 2η²⟩, where λ₀ = −4η⁴. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4p^s over ℛ_a. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

In this note, the relationship between Abelian codes and corresponding Constabelian codes of smaller lengths has been studied.

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.

Let $R_{u^2,v^2,p^k}={𝔽}_{p^k}+u{𝔽}_{p^k}+v{𝔽}_{p^k}+uv{𝔽}_{p^k}$, where $u^2=0,v^2=0$, $uv=vu$, $p$ is a prime and $k$ is a positive integer. We define a gray map from a linear code of length $n$ over the ring $R_{u^2,v^2,p^k}$ to a linear code of length $p^{2k}n$ over the field ${𝔽}_{p^k}$. In this paper, we characterize the gray images of $(1-u)$-constacyclic codes of an arbitrary length over the ring $R_{u^2,v^2,p^k}$ in terms of quasicyclic codes over ${𝔽}_{p^k}$. We obtain some optimal linear codes over ${𝔽}_4$ as gray images.

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that ${𝔽}_q$ is the finite field of order $q,$ where $q$ is a power of the prime $p,$ and $p,\ell$ are distinct odd primes, and $m,n$ are positive integers. We determine generator polynomials of all constacyclic codes of length $8{\ell }^m{p}^n$ over the finite field ${𝔽}_q.$ We also determine their dual codes.

In this paper, structural properties of $(1-2v)$-constacyclic codes over the finite non-chain ring ${𝔽}_q+u{𝔽}_q+v{𝔽}_q+uv{𝔽}_q$ are studied, where $u^2=u$, $v^2=v$, $uv=vu$ and $q$ is a power of some odd prime. As an application, some better quantum codes, compared with previous work, are obtained.

Let $R$ be the ring $F_q[u,v,w]/〈u^2-1,v^2-1,w^3-w,uv-vu,vw-wv,wu-uw〉$, where $q=p^m$ for any odd prime $p$ and positive integer $m$. In this paper, we study constacyclic codes over the ring $R$. We define a Gray map by a matrix and decompose a constacyclic code over the ring $R$ as the direct sum of constacyclic codes over $F_q$, we also characterize self-dual constacyclic codes over the ring $R$ and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring $R$.

Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I m-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II m-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer s to be a multiplier of a Type-II m-adic constacyclic code are determined. Subsequently, for a given positive integer m, a necessary condition and a sufficient condition for the existence of Type-II m-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters.