Sequences of Bivariate Bicycle Codes from Covering Graphs
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-\textit{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 new 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 [[nh = hn, kh, dh]], we prove that kh ≥ k for any h, and dh ≤ hd when h is odd. Furthermore if h is odd and kh = k, we prove the distance lower bound d ≤ dh. We conjecture it is always true that an h-cover BB code of a base $[[n,k,d]]$ BB code obeys the distance bounds d ≤ dh ≤ 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.
Related
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Existence and Characterization of Bivariate Bicycle Codes
- Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
- Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits
- Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
BibTeX
@article{anonymous2026sequences,
title = {{Sequences of Bivariate Bicycle Codes from Covering Graphs}},
author = {Anonymous},
journal = {PRX Quantum},
year = {2026},
month = aug,
doi = {10.1103/vxpb-vjwl},
url = {https://doi.org/10.1103/vxpb-vjwl},
note = {CC BY},
}
RIS
TY - JOUR
TI - Sequences of Bivariate Bicycle Codes from Covering Graphs
AU - Anonymous
PY - 2026
DA - 2026/08/03
JO - PRX Quantum
DO - 10.1103/vxpb-vjwl
UR - https://doi.org/10.1103/vxpb-vjwl
AB - 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-\textit{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 new 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 ≥ k for any h, and d_h ≤ hd when h is odd. Furthermore if h is odd and k_h = k, we prove the distance lower bound d ≤ d_h. We conjecture it is always true that an h-cover BB code of a base $[[n,k,d]]$ BB code obeys the distance bounds d ≤ d_h ≤ 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.
SN - 2691-3399
ER -