Sequences of Bivariate Bicycle Codes from Covering Graphs
Benjamin C. B. Symons, Abhishek Rajput, Dan E. Browne
TL;DR
The paper addresses constructing large BB-code families by covering graphs of Tanner graphs, establishing exact algebraic conditions that guarantee valid $h$-cover BB codes and linking their parameters via $(co)$homology maps.The main approach combines graph-cover theory with BB-code structure, enabling projection and lifting of logical operators and automorphisms across base and cover codes, and derives distance and rate bounds for odd covers.Key findings include a proven bound $k_h \ge k$ for odd $h$, $d_h \le hd$ with potential lower bounds when $k_h=k$, and extensive numerical evidence showing many new and improved BB-codes (including weight-8 codes) arising as cover codes, offering a scalable path toward families of qLDPC codes.The work has practical significance for constructing high-rate, good-distance quantum LDPC codes with manageable search spaces and provides tools to transfer logical structure between base and cover codes, with extensions to automorphisms and potential generalizations to broader code classes.
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.