140 research outputs found

    Minimal Reversible Nonsymmetric Rings

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    Marks showed that F2Q8\mathbb{F}_2Q_8, the F2\mathbb{F}_2 group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although F2Q8\mathbb{F}_2Q_8 is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo

    Duality Preserving Gray Maps for Codes over Rings

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    Given a finite ring AA which is a free left module over a subring RR of AA, two types of RR-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over AA to codes over RR. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance

    Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

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    Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven.Comment: arXiv admin note: text overlap with arXiv:0906.400

    Properties of dual codes defined by nondegenerate forms

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    Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight

    A transform approach to polycyclic and serial codes over rings

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    Producción CientíficaIn this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.Ministerio de Ciencia, Innovación y Universidades / Agencia Estatal de Investigación / 0.13039/501100011033 (grant PGC2018-096446-B-C21

    The use of chemicals in the control of certain herbaceous perennial weeds

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    LD2668 .T4 1954 S99Master of Scienc
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