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On Finite Rings, Algebras, and Error-Correcting Codes

Hieta-aho, Erik
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1525182104493243

Abstract Details

2018, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
This dissertation entails four different topics related to coding theory. While much effort has gone recently into studying codes over ring alphabets (rather than the traditional field alphabets), we change this perspective slightly and focus not necessarily on the algebraic structure of the alphabet but in the ancillary structure of the ambient (the algebraically enhanced environment from which codes are drawn.) Doing so, we determine, for example, that when the ambient is a Frobenius algebra a version of the MacWilliams identities holds between the weight distribution of the appropriate codes and their properly defined duals. We further the consideration of rational power series and sequential codes, which had mostly been studied over field alphabets, to the case when the alphabet is a commutative local ring with nilpotent radical. Perhaps the most striking feature of our study is the introduction of a modified division algorithm that is based not on the degree of polynomials but on their codegree (degree of the lowest degree non-zero term.) As was the case in the initial papers on the subject, we explore criteria to recognize the Kronecker Criterion as well as multivariable rational functions among arbitrary multivariable power series with coefficients in a commutative local ring with nilpotent radical. The duals of sequential codes are the so-called polycyclic codes. We have studied intrinsic notions of duality based on the codegree of polynomials with sights to have a fully intrinsic perspective while studying codes in such ambients. We continue the ongoing explorations to recognize commutative local finite rings. Our study considers the feasibility of extending results in the literature for rings with 16 elements to rings with p4 elements (p an arbitrary prime.)
Sergio Lopez-Permouth (Advisor)
Alexei Davydov (Committee Member)
Adam Fuller (Committee Member)
Jeffery Dill (Committee Member)
106 p.
Mathematics
error correcting codes; Frobenius algebras; finite rings; skew cyclic codes; ambient algebra; local rings; polynomials over finite rings; non degenerate bilinear form; rational functions; power series; sequential codes

Recommended Citations

Citations

  • Hieta-aho, E. (2018). On Finite Rings, Algebras, and Error-Correcting Codes [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1525182104493243

    APA Style (7th edition)

  • Hieta-aho, Erik. On Finite Rings, Algebras, and Error-Correcting Codes. 2018. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1525182104493243.

    MLA Style (8th edition)

  • Hieta-aho, Erik. "On Finite Rings, Algebras, and Error-Correcting Codes." Doctoral dissertation, Ohio University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1525182104493243

    Chicago Manual of Style (17th edition)

ohiou1525182104493243
746
© 2018, all rights reserved.
This open access ETD is published by Ohio University and OhioLINK.