Abstract
Cyclic codes arise naturally as ideals of the commutative ring $F[x]/\langle x^n-1 \rangle$, while a comparable algebraic description for quasi-cyclic (QC) codes has remained elusive. In this work, we introduce a noncommutative framework in which QC codes appear naturally as ideals. By embedding graded polynomial quotient rings into finite Szabo matrix rings, we construct a ring whose left,
Status
Graduate
Department
Mathematics
College
College of Arts and Sciences
Campus
Athens
Faculty Mentor
Sergio Lopez-Permouth
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
FROM ENTANGLED SHIFTS TO QUASI-CYCLIC CODES
Cyclic codes arise naturally as ideals of the commutative ring $F[x]/\langle x^n-1 \rangle$, while a comparable algebraic description for quasi-cyclic (QC) codes has remained elusive. In this work, we introduce a noncommutative framework in which QC codes appear naturally as ideals. By embedding graded polynomial quotient rings into finite Szabo matrix rings, we construct a ring whose left,