Fast Algorithms for the Computation of the Minimum Distance of a Random Linear Code

The minimum distance of a code is an important concept in information theory. Hence, computing the minimum distance of a code with a minimum computational cost is a crucial process to many problems in this area. In this paper, we present and evaluate a family of algorithms and implementations to compute the minimum distance of a random linear code over $\mathbb{F}_{2}$ that are faster than different current implementations. In addition to the basic sequential implementations, we present parallel and vectorized implementations that render high performances on modern architectures. The attained performance results show the benefits of the developed optimized algorithms, which obtain remarkable performance improvements compared with state-of-the-art implementations widely used nowadays.

[1]  Teofilo C. Ancheta.  Syndrome-source-coding and its universal generalization , 1976, IEEE Trans. Inf. Theory.

[2]  Fernando Hernando,et al.  Construction and decoding of matrix-product codes from nested codes , 2009, Applicable Algebra in Engineering, Communication and Computing.

[3]  Robert J. McEliece,et al.  A public key cryptosystem based on algebraic coding theory , 1978 .

[4]  Tom Høholdt,et al.  List decoding of matrix-product codes from nested codes: An application to quasi-cyclic codes , 2012, Adv. Math. Commun..

[5]  Yuan Luo,et al.  Relative generalized Hamming weights of one-point algebraic geometric codes , 2014, 2014 IEEE Information Theory Workshop (ITW 2014).

[6]  Fernando Hernando,et al.  New linear codes from matrix-product codes with polynomial units , 2010, Adv. Math. Commun..

[7]  Fernando Hernando,et al.  Decoding of Matrix-Product Codes , 2011, ArXiv.

[8]  Adi Shamir,et al.  How to share a secret , 1979, CACM.

[9]  Shuo-Yen Robert Li,et al.  Linear network coding , 2003, IEEE Trans. Inf. Theory.

[10]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[11]  Markus Grassl,et al.  Searching for linear codes with large minimum distance , 2006 .

[12]  T. Ho,et al.  On Linear Network Coding , 2010 .

[13]  Zixiang Xiong,et al.  Compression of binary sources with side information at the decoder using LDPC codes , 2002, IEEE Communications Letters.

[14]  Alexander Vardy,et al.  The intractability of computing the minimum distance of a code , 1997, IEEE Trans. Inf. Theory.