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_5: Low-density parity-check (LDPC) codes achieve outstanding performance by the sum-product decoder. Its performance approaches the maximum-likelihood (ML) decoding performance at high SNR region. Upper bounds on the error probability of the ML decoded LDPC codes can be computed using the weight distribution. Recently, X.-Y. presented an approximate algorithm for estimating the weight distribution of LDPC codes. However, the weight distribution computed by their method depends on the iterative decoding algorithm, and is not always estimated exactly. In this paper, we propose a probabilistic method for computing the low-part weight distribution of LDPC codes. Using our method, the approximate weight distribution of LDPC codes can be computed with high accuracy
@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_7: 
@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 .
For a linear code @math , the minimum weight is equivalent to the minimum distance, @math . This is of interest to both coding theorists and cryptographers, and the algorithms used are of a somewhat affine nature to the one here proposed, so we briefly discuss the topic - mostly following the summary in @cite_14 and @cite_11 . In one specific instance @cite_5 , the Stern algorithm for the minimum distance @cite_7 was extended to the estimation of the whole weight distribution of LDPC codes.