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Ring Theory Seminar - Erik Hieta-aho

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February 9, 2018
4:45PM - 5:45PM
Cockins Hall 240

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Add to Calendar 2018-02-09 16:45:00 2018-02-09 17:45:00 Ring Theory Seminar - Erik Hieta-aho Title: Error Correcting Codes in a Frobenius Algebra AmbientSpeaker: Erik Hieta-aho (Ohio University, Athens)Abstract: Cyclic codes are among the most studied error-correcting codes. Negacyclic, constacyclic and polycyclic codes are systematic generalizations of cyclic codes. Their underlying common feature is that they can be considered as ideals of certain rings (their Ambient ring.) Cyclic and negacyclic codes share the appealing property that the dual of a cyclic (negacyclic) code is also cyclic (negacyclic) code; in fact the duals are ideals of the same ambient ring.On the other hand, while Constacyclic codes still satisfy that their duals are of the same type, a constacyclic code and its dual are not necessarily ideals of the same ambient ring. The relationship between such pairs of ambient rings has recently been explored in [2]. Noting the fact that the duals of polycyclic codes are not polycyclic [3] and observing the alternative of using annihilators in lieu of dual codes proposed and studied in [1] suggests an alternative approach.We extend the results in [1] by assuming only that the ambient ring is a Frobenius algebra. While Frobenius rings in general satisfy the double annihilator condition and that makes it so that an ideal is completely determined by its annihilator, we have only been successful so far in the context of a Frobenius algebra where the additional structure has allowed us to construct an appropriate balanced non-degenerate bilinear form. We have also managed to obtain analogues to the MacWilliams identities in this setting. This is a preliminary report on a joint work with José Gómez-Torrecillas, Javier Lobillo, Sergio López-Permouth and Gabriel Navarro.[1] Alahmadi, Dougherty, Leroy, and Solé, On the Duality and the direction of polycyclic codes, Advances in Mathematics of Communications 10, (2016), 923-931.[2] Gómez-Torrecillas, Lobillo, and Navarro. Dual Skew Codes from Annihilators: Transpose Hamming ring extensions, preprint, 2017.[3] López-Permouth, Parra-Avila, and Szabo, Dual generalizations of the concept of cyclicity of codes, Advances in Mathematics of Communications 3, (2009), 227-234. Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public

Title: Error Correcting Codes in a Frobenius Algebra Ambient

Speaker: Erik Hieta-aho (Ohio University, Athens)

Abstract: Cyclic codes are among the most studied error-correcting codes. Negacyclic, constacyclic and polycyclic codes are systematic generalizations of cyclic codes. Their underlying common feature is that they can be considered as ideals of certain rings (their Ambient ring.) Cyclic and negacyclic codes share the appealing property that the dual of a cyclic (negacyclic) code is also cyclic (negacyclic) code; in fact the duals are ideals of the same ambient ring.On the other hand, while Constacyclic codes still satisfy that their duals are of the same type, a constacyclic code and its dual are not necessarily ideals of the same ambient ring. The relationship between such pairs of ambient rings has recently been explored in [2]. Noting the fact that the duals of polycyclic codes are not polycyclic [3] and observing the alternative of using annihilators in lieu of dual codes proposed and studied in [1] suggests an alternative approach.We extend the results in [1] by assuming only that the ambient ring is a Frobenius algebra. While Frobenius rings in general satisfy the double annihilator condition and that makes it so that an ideal is completely determined by its annihilator, we have only been successful so far in the context of a Frobenius algebra where the additional structure has allowed us to construct an appropriate balanced non-degenerate bilinear form. We have also managed to obtain analogues to the MacWilliams identities in this setting. This is a preliminary report on a joint work with José Gómez-Torrecillas, Javier Lobillo, Sergio López-Permouth and Gabriel Navarro.

[1] Alahmadi, Dougherty, Leroy, and Solé, On the Duality and the direction of polycyclic codes, Advances in Mathematics of Communications 10, (2016), 923-931.

[2] Gómez-Torrecillas, Lobillo, and Navarro. Dual Skew Codes from Annihilators: Transpose Hamming ring extensions, preprint, 2017.

[3] López-Permouth, Parra-Avila, and Szabo, Dual generalizations of the concept of cyclicity of codes, Advances in Mathematics of Communications 3, (2009), 227-234.

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