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Ismail Aydogdu - One of the best experts on this subject based on the ideXlab platform.
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Optimal binary Codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$ F 2 F 4 -additive Cyclic Codes
Journal of Applied Mathematics and Computing, 2020Co-Authors: Taher Abualrub, Nuh Aydin, Ismail AydogduAbstract:In this paper, we study the algebraic structure of additive Cyclic Codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ F 2 r × F 4 s = F 2 r F 4 s , where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ F 2 = GF ( 2 ) and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ F 4 = GF ( 4 ) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We also introduce a linear map W that depends on the trace map T to relate these Codes to binary linear Codes over $$\mathbb {F} _{2}.$$ F 2 . Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ F 2 F 4 -additive Cyclic Code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Code. Using the mapping W , we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes whose binary images have optimal parameters. We also consider additive Cyclic Codes over $$\mathbb {F}_{4}$$ F 4 and give some examples of optimal parameter quantum Codes over $$\mathbb {F}_{4}$$ F 4 .
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Optimal binary Codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$F2F4-additive Cyclic Codes
Journal of Applied Mathematics and Computing, 2020Co-Authors: Taher Abualrub, Nuh Aydin, Ismail AydogduAbstract:In this paper, we study the algebraic structure of additive Cyclic Codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ F 2 r × F 4 s = F 2 r F 4 s , where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ F 2 = GF ( 2 ) and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ F 4 = GF ( 4 ) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We also introduce a linear map W that depends on the trace map T to relate these Codes to binary linear Codes over $$\mathbb {F} _{2}.$$ F 2 . Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ F 2 F 4 -additive Cyclic Code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Code. Using the mapping W , we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes whose binary images have optimal parameters. We also consider additive Cyclic Codes over $$\mathbb {F}_{4}$$ F 4 and give some examples of optimal parameter quantum Codes over $$\mathbb {F}_{4}$$ F 4 .
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The structure of ℤ2ℤ2s-additive Cyclic Codes
Discrete Mathematics Algorithms and Applications, 2018Co-Authors: Ismail Aydogdu, Taher AbualrubAbstract:[Formula: see text]-additive Codes for any integer [Formula: see text] are considered as Codes over mixed alphabets. They are a generalization of binary linear Codes and linear Codes over [Formula: see text] In this paper, we are interested in studying [Formula: see text]-additive Cyclic Codes. We will give the generator polynomials of these Codes. We will also give the minimal spanning sets for these Codes. We will define separable [Formula: see text]-additive Codes and provide conditions on the generator polynomials for a [Formula: see text]-additive Cyclic Code to be separable. Finally, we present some examples of optimal parameter binary Codes obtained as images of [Formula: see text]-additive Cyclic Codes.
Joaquim Borges - One of the best experts on this subject based on the ideXlab platform.
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there is exactly one mathbb z _2 mathbb z _4 Cyclic 1 perfect Code
Designs Codes and Cryptography, 2017Co-Authors: Joaquim Borges, Cristina FernandezcordobaAbstract:Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive Code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear Code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-Cyclic Code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this Cyclic Code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect Code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect Code. We also prove that such a Code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-Cyclic.
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There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z
Designs Codes and Cryptography, 2017Co-Authors: Joaquim Borges, Cristina Fernández-córdobaAbstract:Let $$\mathcal{C}$$ C be a $${\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z 2 Z 4 -additive Code of length $$n > 3$$ n > 3 . We prove that if the binary Gray image of $$\mathcal{C}$$ C is a 1-perfect nonlinear Code, then $$\mathcal{C}$$ C cannot be a $${\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z 2 Z 4 -Cyclic Code except for one case of length $$n=15$$ n = 15 . Moreover, we give a parity check matrix for this Cyclic Code. Adding an even parity check coordinate to a $${\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z 2 Z 4 -additive 1-perfect Code gives a $${\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z 2 Z 4 -additive extended 1-perfect Code. We also prove that such a Code cannot be $${\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z 2 Z 4 -Cyclic.
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mathbb z _ 2 mathbb z _ 4 additive Cyclic Codes generator polynomials and dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ 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 $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . 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 ${\mathcal{ C}}$ .
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${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -Additive Cyclic Codes, Generator Polynomials, and Dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernández-córdoba, Roger Ten-vallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ 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 $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . 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 ${\mathcal{ C}}$ .
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z2z4 additive Cyclic Codes generator polynomials and dual Codes
arXiv: Discrete Mathematics, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive Code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ 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 $\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. 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 ${\cal C}$.
Taher Abualrub - One of the best experts on this subject based on the ideXlab platform.
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Optimal binary Codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$ F 2 F 4 -additive Cyclic Codes
Journal of Applied Mathematics and Computing, 2020Co-Authors: Taher Abualrub, Nuh Aydin, Ismail AydogduAbstract:In this paper, we study the algebraic structure of additive Cyclic Codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ F 2 r × F 4 s = F 2 r F 4 s , where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ F 2 = GF ( 2 ) and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ F 4 = GF ( 4 ) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We also introduce a linear map W that depends on the trace map T to relate these Codes to binary linear Codes over $$\mathbb {F} _{2}.$$ F 2 . Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ F 2 F 4 -additive Cyclic Code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Code. Using the mapping W , we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes whose binary images have optimal parameters. We also consider additive Cyclic Codes over $$\mathbb {F}_{4}$$ F 4 and give some examples of optimal parameter quantum Codes over $$\mathbb {F}_{4}$$ F 4 .
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Optimal binary Codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$F2F4-additive Cyclic Codes
Journal of Applied Mathematics and Computing, 2020Co-Authors: Taher Abualrub, Nuh Aydin, Ismail AydogduAbstract:In this paper, we study the algebraic structure of additive Cyclic Codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ F 2 r × F 4 s = F 2 r F 4 s , where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ F 2 = GF ( 2 ) and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ F 4 = GF ( 4 ) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We also introduce a linear map W that depends on the trace map T to relate these Codes to binary linear Codes over $$\mathbb {F} _{2}.$$ F 2 . Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ F 2 F 4 -additive Cyclic Code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Code. Using the mapping W , we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ F 2 F 4 -additive Cyclic Codes whose binary images have optimal parameters. We also consider additive Cyclic Codes over $$\mathbb {F}_{4}$$ F 4 and give some examples of optimal parameter quantum Codes over $$\mathbb {F}_{4}$$ F 4 .
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The structure of ℤ2ℤ2s-additive Cyclic Codes
Discrete Mathematics Algorithms and Applications, 2018Co-Authors: Ismail Aydogdu, Taher AbualrubAbstract:[Formula: see text]-additive Codes for any integer [Formula: see text] are considered as Codes over mixed alphabets. They are a generalization of binary linear Codes and linear Codes over [Formula: see text] In this paper, we are interested in studying [Formula: see text]-additive Cyclic Codes. We will give the generator polynomials of these Codes. We will also give the minimal spanning sets for these Codes. We will define separable [Formula: see text]-additive Codes and provide conditions on the generator polynomials for a [Formula: see text]-additive Cyclic Code to be separable. Finally, we present some examples of optimal parameter binary Codes obtained as images of [Formula: see text]-additive Cyclic Codes.
Roger Tenvalls - One of the best experts on this subject based on the ideXlab platform.
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mathbb z _ 2 mathbb z _ 4 additive Cyclic Codes generator polynomials and dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ 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 $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . 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 ${\mathcal{ C}}$ .
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z2z4 additive Cyclic Codes generator polynomials and dual Codes
arXiv: Discrete Mathematics, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive Code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ 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 $\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. 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 ${\cal C}$.
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z2z4 additive Cyclic Codes generator polynomials and dual Codes
Proceedings of Karatekin Mathematics Days, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A Z₂Z₄-additive Code C ⊆ Zα2 × Zβ₄ is called Cyclic Code if the set of coordinates can be partitioned into two subsets, the set of Z₂ and the set of Z₄ coordinates, such that any Cyclic shift of the coordinates of both subsets leaves invariant the Code. These Codes can be identified as submodules of the Z₄[x]-module Z₂[x]/(x^α − 1) × Z₄ [x]/(x^β − 1). The parameters of a Z₂Z₄-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 Z₂Z₄-additive Cyclic Code are determined in terms of the generator polynomials of the Code C.
Patrick Solé - One of the best experts on this subject based on the ideXlab platform.
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how many weights can a Cyclic Code have
IEEE Transactions on Information Theory, 2020Co-Authors: Minjia Shi, Alessandro Neri, Patrick SoléAbstract:Upper and lower bounds on the largest number of weights in a Cyclic Code of given length, dimension and alphabet are given. An application to irreducible Cyclic Codes is considered. Sharper upper bounds are given for the special Cyclic Codes (called here strongly Cyclic), whose nonzero Codewords have period equal to the length of the Code. Asymptotics are derived on the function $\Gamma (k,q)$ , that is defined as the largest number of nonzero weights a Cyclic Code of dimension $k$ over $\mathbb {F}_{q}$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller Codes, either binary or $q$ -ary, as well as in the $q$ -ary Hamming Code are determined, two difficult results of independent interest.
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How many weights can a Cyclic Code have ?
IEEE Transactions on Information Theory, 2020Co-Authors: Minjia Shi, Alessandro Neri, Patrick SoléAbstract:Upper and lower bounds on the largest number of weights in a Cyclic Code of given length, dimension and alphabet are given. An application to irreducible Cyclic Codes is considered. Sharper upper bounds are given for the special Cyclic Codes (called here strongly Cyclic), {whose nonzero Codewords have period equal to the length of the Code}. Asymptotics are derived on the function $\Gamma(k,q),$ {that is defined as} the largest number of nonzero weights a Cyclic Code of dimension $k$ over $\F_q$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller Codes, either binary or $q$-ary, as well as in the $q$-ary Hamming Code are determined, two difficult results of independent interest.
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How many weights can a quasi-Cyclic Code have ?
IEEE Transactions on Information Theory, 2020Co-Authors: Minjia Shi, Alessandro Neri, Patrick SoléAbstract:We investigate the largest number of nonzero weights of quasi-Cyclic Codes. In particular, we focus on the function ΓQ(n, , k, q), that is defined to be the largest number of nonzero weights a quasi-Cyclic Code of index gcd(, n), length n and dimension k over Fq can have, and connect it to similar functions related to linear and Cyclic Codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a q-ary Reed-Muller Code is quasi-Cyclic, a result of independent interest.
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How many weight can a Cyclicc ode have?
IEEE Transactions Information Th., 2019Co-Authors: Minjia Shi, Alessandro Neri, Patrick SoléAbstract:Upper and lower bounds on the largest number of weights in a Cyclic Code of given length, dimension and alphabet are given. An application to irreducible Cyclic Codes is considered. Sharper upper bounds are given for the special Cyclic Codes (called here strongly Cyclic), whose nonzero Codewords have period equal to the length of the Code. Asymptotics are derived on the function Γ(k, q), that is defined as the largest number of nonzero weights a Cyclic Code of dimension k over F q can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller Codes, either binary or q-ary, as well as in the q-ary Hamming Code are determined, two difficult results of independent interest.
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Notes on Quasi-Cyclic Codes with Cyclic Constituent Codes
Mathematics Across Contemporary Sciences, 2017Co-Authors: Minjia Shi, Yiping Zhang, Patrick SoléAbstract:Quasi-Cyclic Codes are generalizations of the familiar linear Cyclic Codes. By using the results of [4], the authors in [2, 3] showed that a quasi-Cyclic Code $$\mathscr {C}$$ over $$\mathbb {F}_q$$ of length $$\ell m$$ and index $$\ell $$ with m being pairwise coprime to $$\ell $$ and the characteristic of $$\mathbb {F}_q$$ is equivalent to a Cyclic Code if the constituent Codes of $$\mathscr {C}$$ are Cyclic, where q is a prime power and the equivalence is given in [3]. In this paper, we apply an algebraic method to prove that a quasi-Cyclic Code with Cyclic constituent Codes is equivalent to a Cyclic Code. Moreover, the main result (see Theorem 4) includes Proposition 9 in [3] as a special case.
