Coset Lifted Product Codes: Constructing Quantum LDPC Codes from Non-Normal Subgroup Actions

Abstract

We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$, a subgroup $H\le G$ that need not be normal, a protograph $A$ over ${\mathbb{F}}_2[G]$, and a protograph $B$ over ${\mathbb{F}}_2[N_G(H)]$, the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $\rho$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120\le n\le 400$ yields certified codes that exceed the best published value of $Q=k{d}^2/n$ at check weights $6,7$ and $9$, i.e., we obtain $[[360,20,20]]$ at $w=6$, a $[[360,24,20]]$ code at $w=7$, and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$. Our search also produce high-distance codes with parameters $[[168,10,14]],[[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$.