For this week I have implemented a new decoding algorithm for binary linear codes using Grobner

Basis of the ideal associated to the code.

In the binary case I have decided to use the FGLM algorithm from singular to compute the Grobner

Basis, this algorithm is very fast. So first I have to construct the ideal associated to the code.

Then, I create a test-set for the code. And use this one to apply the descent decoding algorithm.

More details about this algorithm : Theory

The process of computing the grobner basis and the test-set is only executed once, and after that, the decoding process is very fast.(Decoding algorithms with pre-computed information)

I have done some tests to analyze the time it takes with the different decoding algorithms and differents types of binary linear codes:

Results of experiments: Testing Algorithm

Also I have documented the functions with examples and more. The code of this algorithm and functions is in the commit:

https://bitbucket.org/vsuaste/coding-theory/commits/4d5c8255eaec112b191eb095873eaf09e5708ed4

With this algorithm I finished algorithms for binary codes. So from now, I have started trying to generalize the algorithms for the case of codes over any finite field.

In this case we expect to gain some speed with a new version of FGLM algorithm implemented specificaly for general case of codes.

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