Lifted-Product QLDPC Codes in the Polynomial Domain

2026-09-01

AI Summary

本文提出了一种基于多项式域的提升乘积量子低密度奇偶校验码构造方法，通过在商环上表示并利用多项式共轭来确保CSS正交性，从而实现有限长度的代码构建。

Abstract

This paper presents a finite-length polynomial-domain formulation of lifted-product quantum low-density parity-check (QLDPC) codes. We formulate the code construction over the quotient ring F 2[D]/(D L + 1), where polynomial base matrices are lifted entrywise to binary circulant blocks. This representation gives a compact algebraic description of the lifted-product parity-check matrices and allows CSS orthogonality to be analyzed before binary expansion. We show that the standard circulant lifting map is compatible with polynomial conjugation, which implies that the resulting binary matrices satisfy the CSS commutation constraint. The construction is illustrated with a constraint length 7, rate 1/2 NASA convolutional code example, and numerical examples are provided from a 3 x 4 polynomial parity-check matrix. The finite-length performance of selected constructed codes is then evaluated over the depolarizing channel using various benchmark decoders. The resulting framework gives a structured method to construct finite-length lifted-product QLDPC codes from small polynomial base matrices.

Problem

• Lifted-Product QLDPC Codes
• Polynomial Domain
• CSS Orthogonality

Innovation

• polynomial-domain
• lifted-product QLDPC codes
• quotient ring
• CSS orthogonality
• binary circulant blocks