Abelian two-block group-algebra code[1–3]

Description

A 2BGA code LP\((a,b)\) whose group \(G\) is Abelian, equivalently a multi-dimensional index-two quasi-cyclic code. Since any finite Abelian group is a direct product of cyclic groups, the group algebra elements \(a\) and \(b\) can be treated as multivariate polynomials, and the family includes GB (one variable), bivariate bicycle (two variables), and multivariate bicycle codes, providing some of the best known short codes with small stabilizer weights.

Writing \(G=\mathbb{Z}_{\ell_1}\times \mathbb{Z}_{\ell_2}\times \cdots \times \mathbb{Z}_{\ell_D}\), the elements \(a\) and \(b\) become \(D\)-variate polynomials in \(\mathbb{F}_q[x_1,\ldots,x_D]/\langle x_1^{\ell_1}-1,\ldots,x_D^{\ell_D}-1 \rangle\), yielding stabilizer generator matrices \begin{align} H_X=(A|B)~,\quad H_Z=(B^T|-A^T)~, \tag*{(1)}\end{align} where \(A\) and \(B\) are Kronecker products of circulant matrices representing \(a\) and \(b\). An equivalent construction in terms of Kronecker products of circulant matrices was introduced in Ref. [1], and related higher-dimensional quasi-cyclic and convolutional quantum codes have been constructed in Ref. [4].

The decomposition of \(G\) into cyclic factors, and hence the number of variables \(D\), is not unique: by the Chinese remainder theorem, a cyclic group \(\mathbb{Z}_{\ell}\) with \(\ell = \prod_i p_i^{k_i}\) splits into a product of coprime cyclic groups, so one may or may not use this decomposition and thereby present the same code with fewer or more variables. For example, \(\mathbb{Z}_{15}\) yields a GB code in one variable, whereas the isomorphic \(\mathbb{Z}_{5}\times\mathbb{Z}_{3}\) yields a bivariate bicycle code in two variables.

In the qubit case, Ref. [5] studied Abelian 2BGA codes built from polynomials in \(r\) variables—there named multivariate bicycle codes—focusing on the trivariate (\(r=3\)) case, whose polynomials feature a third, dependent variable \(z=xy\) supplementing the two variables of a bivariate bicycle code. Examples include weight-five codes, such as a \([[30,4,5]]\) code, that admit bi-planar toric layouts with a single long-range connection per check and depth-seven syndrome-extraction circuits [5]. Such codes with check weight at most six admit bi-planar (thickness-two) Tanner graphs, and there is a criterion for determining whether a code admits a toric layout, in which case all checks are related by translations on a torus; weight-four codes can instead admit tangled toric layouts [5].