Dissertation Defense
On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes, by Benigno Rafael Parra Avila (Ohio University)
| What |
|
|---|---|
| When |
Oct 23, 2009 from 03:00 am to 05:00 pm |
| Where | 320 Morton Hall |
| Contact Name | Benigno Parra |
| Contact Email | parra@math.ohiou.edu |
| Add event to calendar |
vCal
iCal
|
Abstract: Let R be a commutative ring with identity. A power series f in R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Z_m for some positive integer m.
Let F be a field; we prove the equivalence between two notions of rationality in F[[x_1,...,x_n]], and hence in F((x_1,...,x_n)), and thus extend Kronecker's criterion for rationality from F[[x]] to the multivariable setting. We introduce the notion of sequential code natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field, and explore fundamental properties of sequential codes as well as connections with periodicity of sequences and with the related notion of linear recurrence sequences.
A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over F_p for some prime p. We also show that all sequential codes may be obtained by a simple and explicit construction. Using this construction, we get examples of sequential codes which reach certain optimal bonds that cannot be attained by cyclic codes.
Finally, we introduce the notion of polycyclic codes which is another generalization of constacyclicity. We establish a duality between sequentiality and polycyclicity. In particular, it is shown that a code C is sequential and polycyclic if and only if C and its dual C' are both sequential if and only if C and its dual C' are both polycyclic. Furthermore, these equivalent statements characterize the family of constacyclic codes.
