Sequences of Bivariate Bicycle Codes from Covering Graphs

TL;DR

This paper introduces a method to generate infinite sequences of bivariate bicycle (BB) codes using covering graphs, establishing algebraic conditions for code parameters and exploring their properties and potential generalizations.

Contribution

The authors develop a framework for constructing and analyzing cover codes of BB codes via graph coverings, including algebraic conditions and bounds on code parameters.

Findings

Many BB codes can be viewed as cover codes, including the [[144,12,12]] gross code.

New BB codes with weight 8 checks, such as [[64,14,8]] and [[144,14,14]], are identified.

Bounds on parameters of cover codes are established, linking base and cover code properties.

Abstract

We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code's Tanner graph. When a BB code has a Tanner graph that is a $h$-fold covering of the base BB code's Tanner graph, we refer to it as a $h$-cover code. We show that for a BB code to be a $h$-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the $[[144,12,12]]$ gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many codes, including a $[[64,14,8]]$ and $[[144,14,14]]$ code. For an $h$-cover code of an $[[n,k,d]]$ BB code with parameters $[[n_h = hn, k_h, d_h]]$, we prove that $k_h \geq k$ and $d_h \leq hd$ when $h$ is odd. Furthermore if $h$ is odd and $k_h = k$, we prove the lower bound $d \leq d_h$. We conjecture it is always true that an $h$-cover BB code of a base $[[n,k,d]]$ BB code has parameters $[[n_h = hn, k_h \geq k, d \leq d_h \leq hd]]$. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes and to certain code constructions involving hypergraph, lifted, and balanced products.