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<!-- This is an automatically generated file. --> <Chapter Label="Chapter_Introduction"> <Heading>Introduction</Heading> <P/> The &GAP; package &QDistRnd; implements a probabilistic algorithm for finding the distance of a <Math>q</Math>-ary quantum low-density parity-check code linear over a finite field <Math>F=\mathop{\rm GF}(q)</Math>. While there is no guarantee of the performance of the algorithm (the existing bounds in the case of quantum LDPC codes are weak, see <Ref Subsect="Subsection_TheAlgorithmDetails"/>), an empirical convergence criterion is given to estimate the probability that a minimum weight codeword has been found. Versions for CSS and regular stabilizer codes are given, see Section <Ref Sect="Section_DistanceFunctions"/> <P/> In addition, a format for storing matrices associated with <Math>q</Math>-ary quantum codes is introduced and implemented, see Chapter <Ref Chap="Chapter_FileFormat"/> and Sec. <Ref Sect="Section_IOFunctions"/>. The format is based on the well establised MaTrix market eXchange (MTX) Coordinate format developed at NIST, and is designed for full backward compatibility with this format. Thus, the files are readable by any software package which supports MTX. <P/> The routines in the package are derived from the code originally written by one of the authors (LPP). A related Covering Set algorithm has a provable performance for generic (non-LDPC) quantum codes based on random matrices <Cite Key="Dumer-Kovalev-Pryadko-IEEE-2017"/>. Implemented version is a variant of the random <Emph>information set</Emph> (IS) algorithm based on random column permutations and Gauss' elimination <Cite Key="Leon-1988"/> <Cite Key="Kruk-1989"/> <Cite Key="Coffey-Goodman-1990"/>. <P/> The &GAP; computer algebra system was chosen because of its excellent support for linear algebra over finite fields. Here we give a reference implementation of the algorithm, with a focus on matrix formats and generality, as opposed to performance. Nevertheless, the routines are sufficiently fast when dealing with codes of practically important block lengths <Math>n\lesssim 10^3</Math>. <P/> </Chapter> ¤ Dauer der Verarbeitung: 0.14 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28