Computing the distance distribution of systematic non-linear codes

The most important families of non-linear codes are systematic. A brute-force check is the only known method to compute their weight distribution and distance distribution. On the other hand, it outputs also all closest word pairs in the code. In the black-box complexity model, the check is optimal among closest-pair algorithms. In this paper we provide a Groebner basis technique to compute the weight/distance distribution of any systematic non-linear code. Also our technique outputs all closest pairs. Unlike the check, our method can be extended to work on code families.

[1]  Ilaria Simonetti,et al.  Gröbner Bases for the Distance Distribution of Systematic Codes , 2009, Gröbner Bases, Coding, and Cryptography.

[2]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

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

[4]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[5]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[6]  Tom Verhoeff,et al.  An updated table of minimum-distance bounds for binary linear codes , 1987, IEEE Trans. Inf. Theory.

[7]  Hans Schönemann,et al.  SINGULAR: a computer algebra system for polynomial computations , 2001, ACCA.

[8]  Frederik Armknecht,et al.  Efficient Computation of Algebraic Immunity for Algebraic and Fast Algebraic Attacks , 2006, EUROCRYPT.

[9]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[10]  Frederik Armknecht,et al.  Algebraic Attacks on Combiners with Memory , 2003, CRYPTO.

[11]  Bruno Buchberger,et al.  Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal , 2006, J. Symb. Comput..

[12]  Jeffrey S. Leon,et al.  A probabilistic algorithm for computing minimum weights of large error-correcting codes , 1988, IEEE Trans. Inf. Theory.

[13]  Carlo Traverso,et al.  A "Divide and Conquer" Algorithm for Hilbert-Poincaré Series, Multiplicity and Dimension of Monomial Ideals , 1993, AAECC.

[14]  Jean-Charles Faugère,et al.  Complexity of Gröbner basis computation for Semi-regular Overdetermined sequences over F_2 with solutions in F_2 , 2002 .

[15]  Franco P. Preparata.  A Class of Optimum Nonlinear Double-Error-Correcting Codes , 1968, Inf. Control..

[16]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[17]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[18]  R. D. Baker,et al.  On the Preparata and Goethals codes , 1983, IEEE Trans. Inf. Theory.

[19]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[20]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[21]  Nathan Linial,et al.  On metric ramsey-type phenomena , 2003, STOC '03.

[22]  K. Roberts,et al.  Thesis , 2002 .

[23]  Robert Krauthgamer,et al.  The black-box complexity of nearest-neighbor search , 2005, Theor. Comput. Sci..