2BGA-codes

This repository contains the quantum error-correcting codes constructed by Hsiang-Ku Lin and Leonid P. Pryadko for the paper Quantum two-block group algebra codes (arXiv:2306.16400).

Codes

The zip archives nonabelian.zip and abelian.zip contain the complete information about all generated two-block group-algebra (2BGA) codes; please see the paper for the details. For example, the file diswtabelian_wt6wtL2_order50_k18.txt from the file abelian.zip contains information about connected 2BGA codes generated from abelian groups of order 50, with total row weight 6 and 1st group element of weight 2, of dimension k=18. The file contains only one row

  50   5   18   2   [ 2 ]   [ 3, 4, 8 ]

where the parameters are:

• the group order $\ell=50$,
• group number m=5 (in the GAP Small Groups Library),
• code dimension k=18,
• code distance d=2,
• and, finally, the lists of non-identity group element indices, in this case defining group algebra elements $a=1+g_2$ and $b=1+g_3+g_4+g_8$ (according to the default order of group elements as defined in GAP).

We used shell scripts to extract useful information from these zip archives.

Scripts

• The shell script file parafig1to5.sh can be used to produce the data for Figs 1 to 5 in the paper.

• The script file parafig6to9.sh can be used to produce the data for Figs 6 to 9 in the paper.

• The script file kd.sh is used to find the codes whose parameters $[[n,k,d]]$ satisfy the inequality $kd > n$, where the code length $n=2\ell$ is twice the group size $\ell$.

GAP file

The file 2BGA.gap contains two main GAP functions:

• the function checkdata() to verify the distance and the dimension of the codes. It takes three arguments:

  • the group to use
  • the list of indices of non-identity group elements in $a$
  • the list of indices of non-identity group elements in $b$

• The function checkrank() to calculate the ranks of matrices A, B, AB, Hx, Hz, and the parameters $\delta_X$ and $\delta_Z$ defined in the paper. It takes the same three arguments.

The indices of the group elements should correspond exactly to the default ordering of the group as it is defined in the SmallGroups library of GAP.