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New Efficient Scheme Based on Reduction of the Dimension in the Multiple Impulse Method to Find the Minimum Distance of Linear Codes

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In order to find a minimum weight codeword in a linear code, the Multiple Impulse Method uses the Ordered Statistics Decoder of order 3 having a complexity which increases with the code dimension. This paper presents an important improvement of this method by finding a sub code of C of small dimension containing a lowest weight codeword. In the case of Binary Extended Quadratic Residue codes, the proposed technique consists on finding a self invertible permutation σ from the projective special linear group and searching a codeword having the minimum weight in the sub code fixed by σ. The proposed technique gives the exact value of the minimum distance for all binary quadratic residue codes of length less than 223 by using the Multiple Impulse Method on the sub codes in less than one second. For lengths more than 223, the obtained results prove the height capacity of the proposed technique to find the lowest weight in less time. The proposed idea is generalized for BCH codes and it has permits to find the true value of the minimum distance for some codes of lengths 1023 and 2047. The proposed methods performed very well in comparison to previously known results.
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Automorphism Group; Projective Special Linear Group; Quadratic Residue Codes; Minimum Distance; BCH Codes; Designed Distance; Minimum Weight; Multiple Impulse Method

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