Pair-partition CPM code[1]

Description

Qubit CSS code whose \(X\)- and \(Z\)-type stabilizer generator matrices are \(J\)-by-\(L\) arrays of \(P\)-by-\(P\) circulant permutation matrices (CPMs), with the CSS condition enforced by pairing up block columns [1]. Each block of \(H_XH_Z^T\) is a sum of \(L\) CPMs, one per block column. A pair partition, one per block, splits the block columns into \(L/2\) pairs, and the CPM exponents are constrained so that the two CPMs in each pair coincide and cancel over \(\mathbb{F}_2\). CSS orthogonality thereby becomes a homogeneous linear condition on the exponents [1].

Fix a column weight \(J\), an even row weight \(L>J\), and a prime lift size \(P\). Let \(C(s)\) be the \(P\)-by-\(P\) CPM with shift \(s\in\mathbb{F}_P\), and let \((e_{i\ell})\) and \((d_{j\ell})\) be \(J\)-by-\(L\) exponent arrays over \(\mathbb{F}_P\). The generator matrices \(H_X=(C(e_{i\ell}))\) and \(H_Z=(C(d_{j\ell}))\) are \(G\)-lifts of the complete \(J\)-by-\(L\) protograph over the cyclic group of order \(P\), with length \(n=LP\), row weight \(L\), and column weight \(J\). A \(J\)-by-\(J\) array \((M_{ij})\) of pair partitions, one for each \(X\)-type block row \(i\) and \(Z\)-type block row \(j\), is fixed before the exponents. For each pair \(\{u,v\}\in M_{ij}\), the mixed-difference equation \(e_{iu}-d_{ju}=e_{iv}-d_{jv}\) is imposed. It makes the contributions of block columns \(u\) and \(v\) to block \((i,j)\) of \(H_XH_Z^T\) identical. The exponents are solutions of the resulting homogeneous system over \(\mathbb{F}_P\) [1].

Reported instances include the \((3,10)\)-regular \([[2230,896,24]]\) code with \(P=223\) and Tanner girth eight [1; Table I]. See Ref. [1; Table I] for the full list of reported codes.