A statistical estimation algorithm of the weight distribution of a linear code is shown, based on using its generator matrix as a compression function on random bit strings.
@cite_14: An algorithm for finding minimum-weight words in large linear codes is developed. It improves all previous attacks on the public-key cryptosystems based on codes and it notably points out some weaknesses in McEliece's (1978) cipher. We also determine with it the minimum distance of some BCH codes of length 511.
@cite_11: Decoding random linear codes is a fundamental problem in complexity theory and lies at the heart of almost all code-based cryptography. The best attacks on the most prominent code-based cryptosystems such as McEliece directly use decoding algorithms for linear codes. The asymptotically best decoding algorithm for random linear codes of length n was for a long time Stern’s variant of information-set decoding running in time O 20.05563n . Recently, Bernstein, Lange and Peters proposed a new technique called Ball-collision decoding which offers a speed-up over Stern’s algorithm by improving the running time to O 20.05558n . In this paper, we present a new algorithm for decoding linear codes that is inspired by a representation technique due to Howgrave-Graham and Joux in the context of subset sum algorithms. Our decoding algorithm offers a rigorous complexity analysis for random linear codes and brings the time complexity down to O 20.05363n .
The Lee-Brickell algorithm, as well as others, can be interpreted in the context of information set decoding, an equivalent definition of which can be given in terms of the generator matrix or the parity check matrix - see for instance @cite_14 and @cite_11 . with @math the @math th column of @math .