title: README author: - Leonid P. Pryadko - Weilei Zeng

date: 2024-08-01

dist-m4ri - distance of a classical or quantum CSS code

Overview

The program implements two algorithms for calculating the distance of a classical or quantum CSS binary code:

• Random information set (also, random window, or RW), to calculate the upper distance bound, and
• Connected cluster (CC), to calculate the actual distance or lower distance bound of an LDPC code (quantum or classical).

For a classical (binary linear) code, only matrix H, the parity-check matrix, should be specified.

For a quantum CSS code, matrix H=Hx and either G=Hz or L=Lx matrices are needed.

All matrices with entries in GF(2) should have the same number of columns, n, and obey the following orthogonality conditions: $$HXHZ^T=0,\quad HXLZ^T=0,\quad LXHZ^T=0,\quad LXLZ^T=I,$$ where $I$ is an identity matrix. Notice that the latter identity is not required; it is sufficient that Lx and Lz matrices have the same full row rank =k, the dimension of the code, each row of Lx has a non-zero scalar product with a row of Lz, and vice versa.

How it works: RW algorithm (method=1)

Given the error model, i.e., the matrices $H=Hx$, $L=Lx$ ($L$ is empty for a classical code), the program searches for smallest-weight binary codewords $c$ such that $Hc=0$, $Lc\neq0$.

It repeatedly calculates reduced row echelon form of H, with columns taken in random order, which uniquely fixes the information set (non-pivot columns). Generally, column permutation and row reduction gives $H=U\,(I|A)\,P$, where $U$ is invertible, $P$ is a permutation matrix, $I$ is the identity matrix of size given by the rank of $H$, and columns of $A$ form the information set of the corresponding binary code. The corresponding codewords can be drawn as the rows of the matrix $(A^T|I')\,P$. Because of the identity matrix $I'$, the distribution of such vectors is tilted toward smaller weight, which qualitatively explains why it works.

To speed up the distance calculation, you can use the parameter wmin and wmax (by default, wmin=1 and wmax=-1). When non-zero, if a code word of weight w $\le$ wmin is found, the distance calculation is terminated immediately, and the result -w with a negative sign is returned. This is useful, e.g., if we need to construct a code with big enough distance.

Additional command-line parameters relevant for this method:

• steps the number of RW decoding steps (the number of information sets to be constructed).

How it works: CC algorithm (method=2).

The program tries to construct a codeword recursively, by starting with a non-zero bit in a position i in the range from $0$ to $n-1$, where $n$ is the number of columns, and then recursively adding the additional bits in the support of unsatisfied checks starting from the top. The complexity to enumerate all codewords of weight up to $w$ can be estimated as $n\,(\Delta-1)^{w-1}$, where $\Delta$ is the maximum row weight.

Additional command-line parameters relevant for this method:

• wmax the maximum size of the connected cluster.

• start the position to start the cluster. In this case only one starting position i=start will be used. This is useful, e.g., if the code is symmetric (as, e.g., for cyclic codes).

How to run it

For help, just run ./dist_m4ri -h or ./dist_m4ri --help. This shows the following ```sh $ ./distm4ri --help ./distm4ri: distance of a classical or quantum CSS code usage: ./dist_m4ri parameter=value [...]

Required parameter: method=[int]: bitmap for method used (required, default 0: none): 1: random window (RW) algorithm. Options: steps=[int]: how many information sets to use (1) wmin=[int]: minimum distance of interest (1) 2: connected cluster (CC) algorithm. Options: wmax=[int]: maximum cluster weight (5) start=[int]: use only this position to start (-1)

General parameters: finH=[str]: parity check matrix Hx (NULL) finG=[str]: matrix Hz (quantum CSS code only) (NULL) finL=[str]: matrix Lx (quantum CSS code only) (NULL) Either L=Lx or G=Hz matrix is required for a quantum CSS code fin=[str]: base name for input files ("try") set finH->"${fin}X.mtx" finG->"${fin}Z.mtx" css=[int]: reserved for future use (1) seed=[int]: rng seed use 0 for time(NULL) debug=[int]: bitmap for aux information to output (3) 0: clear the entire debug bitmap to 0. 1: output misc general info (on by default) 2: output more general info (on by default) 4: debug command line arguments parsing 8: output progress reports every 1000 steps 16: output new min-weight codewords found (cut large vectors) 32: output matrices (unless n is large) 64: reserved 128: reserved 256: print out neighbor lists 512: print out vectors/syndrome weights during recursion 1024: print piv/skip_pivs/reserved 2048: allow big matrix / large vector output see the source code for more options Multiple 'debug' parameters are XOR combined except for 0. Use debug=0 as the 1st argument to suppress all debug messages. -h gives this help (also '--help') ```

Compilation

The program is intended for use with recent gcc compilers under linux. Download the distribution from github then run from the dist-m4ri/src directory sh make -j all This should compile the executable dist_m4ri.

The program uses m4ri library for binary linear algebra. To install under Ubuntu, run sudo apt-get update -y sudo apt-get install -y libm4ri-dev