Author Description

Login to generate an author description

Ask a Question About This Mathematician

In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes … In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $\mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $\mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. … In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. Moreover, we investigate Z2Z4Z8-cyclic codes giving their generator polynomials and spanning sets. We also give some illustrative examples of both Z2Z4Z8-additive codes and Z2Z4Z8-cyclic codes.
Inspired by the Z2Z4-additive codes, linear codes over Z_{2}^r x (Z_{2} + uZ_{2})^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to … Inspired by the Z2Z4-additive codes, linear codes over Z_{2}^r x (Z_{2} + uZ_{2})^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z_{2}^r x (Z_{2} + uZ_{2})^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the … In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\mathbb{Z}_8,$ and for additive $\mathbb{Z}_2\mathbb{Z}_4$ codes. Also, we conclude by some properties and examples of the MGN numbers that provide a good source for new number sequences that are not listed in The On-Line Encyclopedia of Integer Sequences.
In this paper we introduce self-dual cyclic and quantum codes over Z2^{\alpha} x (Z2 + uZ2)^{\beta}. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C … In this paper we introduce self-dual cyclic and quantum codes over Z2^{\alpha} x (Z2 + uZ2)^{\beta}. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C^{\perp}. Since the binary image of a self-orthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Z2^{\alpha} x (Z2 + uZ2)^{\beta}. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear … Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and … Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and s positive integers. In this study, we define a new family of codes over the set Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] where \xi is the root of a monic basic primitive polynomial in Z4[x]. We give the standard form of the generator and parity-check matrices of codes over Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] and also we introduce skew cyclic codes and their spanning sets over this set.
In this paper we introduce self-dual cyclic and quantum codes over Z2^α x (Z2 + uZ2)^β. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C … In this paper we introduce self-dual cyclic and quantum codes over Z2^α x (Z2 + uZ2)^β. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C^{\perp}. Since the binary image of a self-orthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Z2^α x (Z2 + uZ2)^β. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.
Additive codes were first introduced by Delsarte in 1973 as subgroups of the underlying abelian group in a translation association scheme. In the case where the association scheme is the … Additive codes were first introduced by Delsarte in 1973 as subgroups of the underlying abelian group in a translation association scheme. In the case where the association scheme is the Hamming scheme, that is, when the underlying abelian group is of order 2 n , the additive codes are of the form Z 2 α × Z 4 β with α + 2 β = n . In 2010, Borges et al. introduced Z 2 Z 4 -additive codes which they defined them as the subgroups of Z 2 α × Z 4 β . In this chapter we introduce Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes where Z 2 = 0 1 is the binary field and Z 2 u = 0 1 u 1 + u is the ring with four elements and u 2 = 0 . We give the standard forms of the generator and parity-check matrices of Z 2 Z 2[u]-linear codes. Further, we determine the generator polynomials for Z 2 Z 2 u -linear cyclic codes. We also present some examples of Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes.
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear … Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and … Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and s positive integers. In this study, we define a new family of codes over the set Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] where \xi is the root of a monic basic primitive polynomial in Z4[x]. We give the standard form of the generator and parity-check matrices of codes over Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] and also we introduce skew cyclic codes and their spanning sets over this set.
In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. … In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. Moreover, we investigate Z2Z4Z8-cyclic codes giving their generator polynomials and spanning sets. We also give some illustrative examples of both Z2Z4Z8-additive codes and Z2Z4Z8-cyclic codes.
Additive codes were first introduced by Delsarte in 1973 as subgroups of the underlying abelian group in a translation association scheme. In the case where the association scheme is the … Additive codes were first introduced by Delsarte in 1973 as subgroups of the underlying abelian group in a translation association scheme. In the case where the association scheme is the Hamming scheme, that is, when the underlying abelian group is of order 2 n , the additive codes are of the form Z 2 α × Z 4 β with α + 2 β = n . In 2010, Borges et al. introduced Z 2 Z 4 -additive codes which they defined them as the subgroups of Z 2 α × Z 4 β . In this chapter we introduce Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes where Z 2 = 0 1 is the binary field and Z 2 u = 0 1 u 1 + u is the ring with four elements and u 2 = 0 . We give the standard forms of the generator and parity-check matrices of Z 2 Z 2[u]-linear codes. Further, we determine the generator polynomials for Z 2 Z 2 u -linear cyclic codes. We also present some examples of Z 2 Z 2[u]-linear and Z 2 Z 2[u]-cyclic codes.
In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes … In this paper we generalize $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-linear codes to codes over $\mathbb{Z}_{p}[u]/{\langle u^r \rangle}\times\mathbb{Z}_{p}[u]/{\langle u^s \rangle}$ where $p$ is a prime number and $u^r=0=u^s$. We will call these family of codes as $\mathbb{Z}_{p}[u^r,u^s]$-linear codes which are actually special submodules. We determine the standard forms of the generator and parity-check matrices of these codes. Furthermore, for the special case $p=2$, we define a Gray map to explore the binary images of $\mathbb{Z}_{2}[u^r,u^s]$-linear codes. Finally, we study the structure of self-dual $\mathbb{Z}_{2}[u^2,u^3]$-linear codes and present some examples.
Inspired by the Z2Z4-additive codes, linear codes over Z_{2}^r x (Z_{2} + uZ_{2})^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to … Inspired by the Z2Z4-additive codes, linear codes over Z_{2}^r x (Z_{2} + uZ_{2})^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z_{2}^r x (Z_{2} + uZ_{2})^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
In this paper we introduce self-dual cyclic and quantum codes over Z2^{\alpha} x (Z2 + uZ2)^{\beta}. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C … In this paper we introduce self-dual cyclic and quantum codes over Z2^{\alpha} x (Z2 + uZ2)^{\beta}. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C^{\perp}. Since the binary image of a self-orthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Z2^{\alpha} x (Z2 + uZ2)^{\beta}. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.
Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and … Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and s positive integers. In this study, we define a new family of codes over the set Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] where \xi is the root of a monic basic primitive polynomial in Z4[x]. We give the standard form of the generator and parity-check matrices of codes over Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] and also we introduce skew cyclic codes and their spanning sets over this set.
In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. … In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. Moreover, we investigate Z2Z4Z8-cyclic codes giving their generator polynomials and spanning sets. We also give some illustrative examples of both Z2Z4Z8-additive codes and Z2Z4Z8-cyclic codes.
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear … Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
In this paper we introduce self-dual cyclic and quantum codes over Z2^α x (Z2 + uZ2)^β. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C … In this paper we introduce self-dual cyclic and quantum codes over Z2^α x (Z2 + uZ2)^β. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C^{\perp}. Since the binary image of a self-orthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Z2^α x (Z2 + uZ2)^β. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.
Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear … Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and … Z2Z4-additive codes have been defined as a subgroup of Z2^{r} x Z4^{s} in [5] where Z2, Z4 are the rings of integers modulo 2 and 4 respectively and r and s positive integers. In this study, we define a new family of codes over the set Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] where \xi is the root of a monic basic primitive polynomial in Z4[x]. We give the standard form of the generator and parity-check matrices of codes over Z2^{r}[\bar{\xi}] x Z4^{s}[\xi] and also we introduce skew cyclic codes and their spanning sets over this set.
In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. … In this paper we study Z2Z4Z8-additive codes, which are the extension of recently introduced Z2Z4-additive codes. We determine the standard forms of the generator and parity-check matrices of Z2Z4Z8-additive codes. Moreover, we investigate Z2Z4Z8-cyclic codes giving their generator polynomials and spanning sets. We also give some illustrative examples of both Z2Z4Z8-additive codes and Z2Z4Z8-cyclic codes.
In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the … In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\mathbb{Z}_8,$ and for additive $\mathbb{Z}_2\mathbb{Z}_4$ codes. Also, we conclude by some properties and examples of the MGN numbers that provide a good source for new number sequences that are not listed in The On-Line Encyclopedia of Integer Sequences.
A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^α\times{\mathbb{Z}}_4^β$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such … A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^α\times{\mathbb{Z}}_4^β$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^α-1)\times\mathbb{Z}_4[x]/(x^β-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic code are determined in terms of the generator polynomials of the code ${\cal C}$.
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.
Several constructions are presented by which spherical codes are generated from groups of binary codes. In the main family of constructions the codes are generated from equally spaced symmetric pointsets … Several constructions are presented by which spherical codes are generated from groups of binary codes. In the main family of constructions the codes are generated from equally spaced symmetric pointsets on the real line. The main ideas are code concatenation and set partitioning. Extensive tables are presented for spherical codes in dimension n/spl les/24.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>
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 determine completely the structure of linear codes over $\mathbb Z/N\mathbb Z$ of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, … In this paper we determine completely the structure of linear codes over $\mathbb Z/N\mathbb Z$ of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.
We introduce a generalization to Z/sub 2/k of the Gray map and generalized versions of Kerdock and Delsarte-Goethals codes. We introduce a generalization to Z/sub 2/k of the Gray map and generalized versions of Kerdock and Delsarte-Goethals codes.
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such … A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n = 1 - 2H(2t/n) where H(p) is the binary entropy function -p log p - (1-p) log (1-p). Upper bounds on this asymptotic rate are given.
We study self-dual codes over the ring Z/sub 2k/ of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce … We study self-dual codes over the ring Z/sub 2k/ of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z/sub 2k/ which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over Z/sub 6/ of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z/sub 2k/ are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z/sub 2k/ and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z/sub 2k/ are also introduced.
We derive theoretical upper and lower bounds on the maximum size of DNA codes of length $n$ with constant GC-content $w$ and minimum Hamming distance $d$, both with and without … We derive theoretical upper and lower bounds on the maximum size of DNA codes of length $n$ with constant GC-content $w$ and minimum Hamming distance $d$, both with and without the additional constraint that the minimum Hamming distance between any codeword and the reverse-complement of any codeword be at least $d$. We also explicitly construct codes that are larger than the best previously-published codes for many choices of the parameters $n$, $d$ and $w$.
In this paper, we study the structure of cyclic codes of an arbitrary length n over the ring Z_2+uZ_2+u^2Z_2+\ldots+u^{k-1}Z_2, where u^k=0. Also we study the rank for these codes, and … In this paper, we study the structure of cyclic codes of an arbitrary length n over the ring Z_2+uZ_2+u^2Z_2+\ldots+u^{k-1}Z_2, where u^k=0. Also we study the rank for these codes, and we find their minimal spanning sets. This study is a generalization and extension of the work in reference [1].
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, 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 $\mathbb{Z}_{q}R$.
We establish some new properties and identities of Generalized Gaussian Numbers (GGN) which are defined recently in [10, 11] parallel to those of Gaussian coefficients. We present generating functions and … We establish some new properties and identities of Generalized Gaussian Numbers (GGN) which are defined recently in [10, 11] parallel to those of Gaussian coefficients. We present generating functions and some properties which are very useful for GGN. We obtain some family of sequences which are unimodal and present the log-concavity property of GGN. Finally, we give a connection of GGN to the Rogers-Szego polynomials.
The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum … The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than 1 - 2 H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).
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.
In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the … In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\mathbb{Z}_8,$ and for additive $\mathbb{Z}_2\mathbb{Z}_4$ codes. Also, we conclude by some properties and examples of the MGN numbers that provide a good source for new number sequences that are not listed in The On-Line Encyclopedia of Integer Sequences.
A code ${\cal C}$ is $\Z_2\Z_4$-additive if the set of coordinates can be partitioned into two subsets $X$ and $Y$ such that the punctured code of ${\cal C}$ by deleting … A code ${\cal C}$ is $\Z_2\Z_4$-additive if the set of coordinates can be partitioned into two subsets $X$ and $Y$ such that the punctured code of ${\cal C}$ by deleting the coordinates outside $X$ (respectively, $Y$) is a binary linear code (respectively, a quaternary linear code). In this paper $\Z_2\Z_4$-additive codes are studied. Their corresponding binary images, via the Gray map, are $\Z_2\Z_4$-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity check matrices are given. For this, the appropriate inner product is deduced and the concept of duality for $\Z_2\Z_4$-additive codes is defined. Moreover, the parameters of the dual codes are computed. Finally, some conditions for self-duality of $\Z_2\Z_4$-additive codes are given.
In this paper, we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a generalisation of cyclic codes … In this paper, we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a generalisation of cyclic codes but constructed using a non-commutative ring called the skew polynomial ring. In previous works, these codes have been studied with certain restrictions on their length. This work examines their structure for an arbitrary length without any restriction. Our results show that these codes are equivalent to either cyclic codes or quasi-cyclic codes, hence establish strong connections with well-known classes of codes.