Spectral Theory of Semisimple Bivariate Bicycle Codes
Information Theory
Summary
The authors extend the classic theory of two-dimensional cyclic codes by creating a new algebraic framework for bivariate bicycle codes. They use special mathematical tools called Frobenius-orbit idempotents to find formulas that determine important code properties like logical size and error tolerance. The authors also develop a systematic way to understand symmetries in these codes to build structured permutations that help in code construction. Their work includes concrete examples that show how to make these codes from basic principles without trial-and-error. Additionally, they broaden their approach to include BCH-based product codes in an appendix.
two-dimensional cyclic codesbivariate bicycle codesFrobenius-orbit idempotentslogical dimensionminimum distancecode symmetriesblock-monomial permutationsBCH codesproduct codes
Authors
Eric Sabo, Mahir Bilen Can, David Marquis
Abstract
Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.