In this paper, we first study the skew cyclic codes of length $ p^s $ over $ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $ where $ p $ is a prime number and …
In this paper, we first study the skew cyclic codes of length $ p^s $ over $ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $ where $ p $ is a prime number and $ u^3 = 0. $ Then we characterize the algebraic structure of $ \mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}[u^2] $-additive skew cyclic codes of length $ 2p^s. $ We will show that there are sixteen different types of these codes and classify them in terms of their generators.
In this paper, we first study the skew cyclic codes of length $ p^s $ over $ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $ where $ p $ is a prime number and …
In this paper, we first study the skew cyclic codes of length $ p^s $ over $ R_3 = \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}, $ where $ p $ is a prime number and $ u^3 = 0. $ Then we characterize the algebraic structure of $ \mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}[u^2] $-additive skew cyclic codes of length $ 2p^s. $ We will show that there are sixteen different types of these codes and classify them in terms of their generators.
Skew polynomial rings over finite fields and over Galois ringshave recently been used to study codes. In this paper, we extend this concept to finite chain rings.Properties of skew constacyclic …
Skew polynomial rings over finite fields and over Galois ringshave recently been used to study codes. In this paper, we extend this concept to finite chain rings.Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is aunit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of suchcodes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of allskew constacyclic codes is completely determined. This allows us to express the generators ofEuclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of theoriginal codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ andtheir Euclidean and Hermitian dual codes is also provided.
We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left …
We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over $GR(4, 2)$ are constructed. Euclidean self-dual codes give self-dual $\mathbb Z_4$−codes. Hermitian self-dual codes yield 3−modular lattices and quasi-cyclic self-dual $\mathbb Z_4$−codes.
Journal Article THE DETERMINATION OF THE GROUP OF AUTOMORHISMS OF A FINITE CHAIN RING OF CHARACTERSTIC p Get access YOUSIF ALKHAMEES YOUSIF ALKHAMEES Mathematics Department Faculty of Science Saud UniversityRiyadh …
Journal Article THE DETERMINATION OF THE GROUP OF AUTOMORHISMS OF A FINITE CHAIN RING OF CHARACTERSTIC p Get access YOUSIF ALKHAMEES YOUSIF ALKHAMEES Mathematics Department Faculty of Science Saud UniversityRiyadh 11451 P.O.Box 2455 Saudi Arabia Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 42, Issue 1, 1991, Pages 387–391, https://doi.org/10.1093/qmath/42.1.387 Published: 01 January 1991 Article history Received: 26 September 1989 Published: 01 January 1991
This paper considers a new alphabet set, which is a ring that we call $\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in …
This paper considers a new alphabet set, which is a ring that we call $\mathbb{F}_4R$, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize $\mathbb{F}_4R$-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over $\mathbb{F}_4$ are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of $R$-skew cyclic codes which are reversible complement.
We study all constacyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over GR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(Rfr,m</i> ), the Galois extension ring of dimension m of the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rfr=</i> …
We study all constacyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over GR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(Rfr,m</i> ), the Galois extension ring of dimension m of the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rfr=</i> F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> +uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . The units of the ring GR <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(Rfr,m</i> ) are of the forms <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha</i> , and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha+u</i> beta, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha,</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">beta</i> are nonzero elements of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> m, which correspond to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> -1) such constacyclic codes. First, the structure and Hamming distances of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1+u</i> gamma)-constacyclic codes are established. We then classify all cyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GR(Rfr,m</i> ), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">alpha</i> -constacyclic codes, as well as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1+u</i> gamma)-constacyclic and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(alpha+u</i> beta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1+u</i> gamma)-constacyclic accordingly to all constacyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GR(Rfr,m</i> ).
Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over …
Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over the alphabet Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> , and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over Galois rings. The structure of negacyclic codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over the Galois rings GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> ,m), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> ,m) in general, and over Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sup> of length 4,8, and 16 are given. The weight distributions of such negacyclic codes are also discussed
In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0.$ We give the definition of these codes as subsets …
In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0.$ We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\Theta]$ are discussed, where $ \Theta$ is an automorphism of $R.$ We describe the generator polynomials of skew constacyclic codes over $\mathbb{Z}_{q}R,$ also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_{4}$. Finally, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.
Constacyclic codes of length <TEX>$p^s$</TEX> over <TEX>$R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$</TEX> are precisely the ideals of the ring <TEX>$\frac{R[x]}{</TEX><TEX><</TEX><TEX>x^{p^s}-1</TEX><TEX>></TEX><TEX>}$</TEX>. In this paper, we investigate constacyclic codes of length <TEX>$p^s$</TEX> over R. The units of …
Constacyclic codes of length <TEX>$p^s$</TEX> over <TEX>$R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$</TEX> are precisely the ideals of the ring <TEX>$\frac{R[x]}{</TEX><TEX><</TEX><TEX>x^{p^s}-1</TEX><TEX>></TEX><TEX>}$</TEX>. In this paper, we investigate constacyclic codes of length <TEX>$p^s$</TEX> over R. The units of the ring R are of the forms <TEX>${\gamma}$</TEX>, <TEX>${\alpha}+u{\beta}$</TEX>, <TEX>${\alpha}+u{\beta}+u^2{\gamma}$</TEX> and <TEX>${\alpha}+u^2{\gamma}$</TEX>, where <TEX>${\alpha}$</TEX>, <TEX>${\beta}$</TEX> and <TEX>${\gamma}$</TEX> are nonzero elements of <TEX>$\mathbb{F}_{p^m}$</TEX>. We obtain the structures and Hamming distances of all (<TEX>${\alpha}+u{\beta}$</TEX>)-constacyclic codes and (<TEX>${\alpha}+u{\beta}+u^2{\gamma}$</TEX>)-constacyclic codes of length <TEX>$p^s$</TEX> over R. Furthermore, we classify all cyclic codes of length <TEX>$p^s$</TEX> over R, and by using the ring isomorphism we characterize <TEX>${\gamma}$</TEX>-constacyclic codes of length <TEX>$p^s$</TEX> over R.
In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial …
In this paper, we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a noncommutative ring called the skew polynomial ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿]. After a brief description of the skew polynomial ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿], it is shown that skew QC codes are left submodules of the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Rsl</i> =( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿]/( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xs</i> -1) ) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> . The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ;¿] and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters.
In this paper, we study skew cyclic codes over the ring R = Fq + uFq + vFq + uvFq, where u 2 = u, v 2 = v, uv …
In this paper, we study skew cyclic codes over the ring R = Fq + uFq + vFq + uvFq, where u 2 = u, v 2 = v, uv = vu, q = p m and p is an odd prime.We investigate the structural properties of skew cyclic codes over R through a decomposition theorem.Furthermore, we give a formula for the number of skew cyclic codes of length n over R.
In this paper, a new class of skew-cyclic codes, termed as F3R-skew cyclic codes, is defined, where R denotes the ring F3 + uF3, u2 = u. These codes are …
In this paper, a new class of skew-cyclic codes, termed as F3R-skew cyclic codes, is defined, where R denotes the ring F3 + uF3, u2 = u. These codes are of length α + β and are submodules of the R-module F3αRβ. We study some structural properties of skew cyclic codes over R and F3R-skew cyclic codes. The generator polynomials of these codes are studied. Some examples are given to illustrate the results. An optimal ternary code is obtained as the Gray image of an F3R-skew cyclic code.