35 research outputs found

    Symplectic self-orthogonal quasi-cyclic codes

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    In this paper, we obtain sufficient and necessary conditions for quasi-cyclic codes with index even to be symplectic self-orthogonal. Then, we propose a method for constructing symplectic self-orthogonal quasi-cyclic codes, which allows arbitrary polynomials that coprime xnβˆ’1x^{n}-1 to construct symplectic self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic codes, we provide lower and upper bounds on the minimum symplectic distances of a class of 1-generator quasi-cyclic codes and their symplectic dual codes. Finally, we construct many binary symplectic self-orthogonal codes with excellent parameters, corresponding to 117 record-breaking quantum codes, improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes. http://www.codetables.de)

    On the equivalence of linear cyclic and constacyclic codes

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    We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Moreover, we provide new algebraic criteria for the monomial equivalence of constacyclic codes over F4\mathbb{F}_4. Finally, we prove that if gcd⁑(3n,Ο•(3n))=1\gcd(3n,\phi(3n))=1, then all permutation equivalent constacyclic codes of length nn over F4\mathbb{F}_4 are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes.Comment: 18 page

    Quantum Codes from additive constacyclic codes over a mixed alphabet and the MacWilliams identities

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    Let Zp\mathbb{Z}_p be the ring of integers modulo a prime number pp where pβˆ’1p-1 is a quadratic residue modulo pp. This paper presents the study of constacyclic codes over chain rings R=Zp[u]⟨u2⟩\mathcal{R}=\frac{\mathbb{Z}_p[u]}{\langle u^2\rangle} and S=Zp[u]⟨u3⟩\mathcal{S}=\frac{\mathbb{Z}_p[u]}{\langle u^3\rangle}. We also study additive constacyclic codes over RS\mathcal{R}\mathcal{S} and ZpRS\mathbb{Z}_p\mathcal{R}\mathcal{S} using the generator polynomials over the rings R\mathcal{R} and S,\mathcal{S}, respectively. Further, by defining Gray maps on R\mathcal{R}, S\mathcal{S} and ZpRS,\mathbb{Z}_p\mathcal{R}\mathcal{S}, we obtain some results on the Gray images of additive codes. Then we give the weight enumeration and MacWilliams identities corresponding to the additive codes over ZpRS\mathbb{Z}_p\mathcal{R}\mathcal{S}. Finally, as an application of the obtained codes, we give quantum codes using the CSS construction.Comment: 22 page

    Quasi-cyclic Hermitian construction of binary quantum codes

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    In this paper, we propose a sufficient condition for a family of 2-generator self-orthogonal quasi-cyclic codes with respect to Hermitian inner product. Supported in the Hermitian construction, we show algebraic constructions of good quantum codes. 30 new binary quantum codes with good parameters improving the best-known lower bounds on minimum distance in Grassl's code tables \cite{Grassl:codetables} are constructed

    New Quantum and LCD Codes over Finite Fields of Even Characteristic

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    For an integer m β‰₯ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m βˆ’ image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m . &nbsp

    Generating generalized necklaces and new quasi-cyclic codes

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    In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13)

    Z_q(Z_q+uZ_q)-Linear Skew Constacyclic Codes

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    In this paper, we study skew constacyclic codes over the ring ZqR\mathbb{Z}_{q}R where R=Zq+uZqR=\mathbb{Z}_{q}+u\mathbb{Z}_{q}, q=psq=p^{s} for a prime pp and u2=0.u^{2}=0. We give the definition of these codes as subsets of the ring ZqαRβ\mathbb{Z}_{q}^{\alpha}R^{\beta}. Some structural properties of the skew polynomial ring R[x,Θ] R[x,\Theta] are discussed, where Θ \Theta is an automorphism of R.R. We describe the generator polynomials of skew constacyclic codes over ZqR,\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 ZqR\mathbb{Z}_{q}R we obtained some new linear codes over Z4\mathbb{Z}_{4}. Finally, we have generalized these codes to double skew constacyclic codes over ZqR\mathbb{Z}_{q}R
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