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. The family includes GB (one variable), bivariate bicycle (two variables), and higher-variable 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 sums of 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]. Related higher-dimensional quasi-cyclic and convolutional quantum codes are constructed in Ref. [4].

The decomposition of \(G\) into cyclic factors, and hence the number of variables \(D\), is not unique (see the MM code entry). 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.

Qubit Abelian 2BGA codes in \(r\) variables have been studied in the trivariate (\(r=3\)) case, in which a third, dependent variable \(z=xy\) supplements the two variables of a bivariate bicycle code [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 [5]. Weight-five examples, such as a \([[30,4,5]]\) code, admit bi-planar toric layouts with a single long-range connection per check and depth-seven syndrome-extraction circuits, whereas weight-four codes can instead admit tangled toric layouts [5].