Bounded distance decoding of linear error-correcting codes with Gröbner bases

In: Journal of Symbolic Computation Special Issue Gröbner Bases Techniques in Cryptography and Coding Theory, to appear. Available online: http://dx.doi.org/10.1016/j.jsc.2007.12.003, 2009

Authors

• Stanislav Bulygin
• Ruud Pellikaan

Abstract

The problem of bounded distance decoding of arbitrary linear codes with the use of Gröbner bases is addressed. A new method is proposed, which is based on reducing an initial decoding problem to solving some system of polynomial equations over a finite field.The peculiarity of this system is that, when we want to decode up to half the minimum distance, it has a unique solution even over the algebraic closure of the considered finite field, although field equations are not added. The equations in the system have degree at most 2. Our method is much faster than the one of Fitzgerald-Lax, which is the only known alternative that decodes with Gröbner bases in this generality. It is also shown via experiments that the proposed approach in some range of parameters is superior to the generic syndrome decoding.

BibTeX

 
@Article{ Bulygin.Pellikaan07bounded, 
   title = { Bounded distance decoding of linear error-correcting codes with Gröbner bases }, 
   author = { Stanislav Bulygin and Ruud Pellikaan },    
   journal = { Journal of Symbolic Computation Special Issue Gröbner Bases Techniques in Cryptography and Coding Theory, to appear },           
   note = { Available online: http://dx.doi.org/10.1016/j.jsc.2007.12.003 },  
   year = 2009, 
} 

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This publication belongs to the project KryFoVe.