Construction of a Family of Quantum Codes Using Sub-exceding Functions via the Hypergraph Product and the Generalized Shor Construction

TL;DR

This paper presents a new scalable family of quantum LDPC codes constructed using sub-exceding functions, hypergraph products, and a generalized Shor method, with promising structural and asymptotic properties.

Contribution

It introduces a novel construction of quantum LDPC codes combining classical codes from sub-exceding functions with hypergraph and Shor techniques, advancing quantum code design.

Findings

Quantum codes have parameters $[[6k^2, k^2, d]]$ with $d=3$ for $k=3$ and $d=4$ for $k extgreater=4$.

Codes exhibit LDPC structure, locality, and promising asymptotic behavior.

The framework enables structured quantum code optimization.

Abstract

In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes $L_k$ and $L_k^{+}$, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters $[[6k^2,\, k^2,\, d]]$. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies $d=3$ for $k=3$ and $d=4$ for $k\ge4$, establishing a new framework for structured quantum LDPC code design and optimization.