Cornucopia code[1]

Description

GALA code with rate just above one half whose \(G\)-lift entries are affine permutations of a \(\mathbb{Z}_3\times\mathbb{Z}_q\) grid. The layout and the syndrome-extraction schedule are designed together, so that a full syndrome cycle takes twelve entangling layers regardless of the code size.

The code has \(n=36q\) qubits partitioned into twelve blocks of \(P=3q\) with \(\gcd(q,3)=1\), each block laid out as a \(3\times q\) array, and three blocks of checks of each type. The twelve lift entries \(A_k\) and \(B_k\), for \(k\in\mathbb{Z}_6\), each act on the grid as an affine map \((x,y)\mapsto(ax+b,y+s)\), so the row part lies in the affine group of \(\mathbb{Z}_3\), which is isomorphic to \(S_3\), while the column part is a cyclic shift. Two entries invert the row coordinate and two translate it, while the remaining eight are pure column shifts. Among the pairs \((A_i,B_j)\) that enter the orthogonality condition, only \((A_1,B_2)\) and \((A_0,B_3)\) fail to commute, and both have \(i+j\equiv3 \bmod 6\), an offset that the three retained check rows never realize [1]. The twelve column shifts are left unconstrained by orthogonality and are the free parameters of the code search.