Determining the generalized hamming weight hierarchy of the binary projective Reed-Muller code
V Ramkumar, M Vajha… - 2018 Twenty Fourth …, 2018 - ieeexplore.ieee.org
2018 Twenty Fourth National Conference on Communications (NCC), 2018•ieeexplore.ieee.org
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the
polynomials being evaluated to yield codewords, are restricted to be homogeneous. The
Generalized Hamming Weights (GHW) of a code C, identify for each dimension v, the
smallest size of the support of a sub code of C of dimension u, The GHW of a code are of
interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use
in bounding the state complexity of the trellis representation of the code. In prior work [1] by …
polynomials being evaluated to yield codewords, are restricted to be homogeneous. The
Generalized Hamming Weights (GHW) of a code C, identify for each dimension v, the
smallest size of the support of a sub code of C of dimension u, The GHW of a code are of
interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use
in bounding the state complexity of the trellis representation of the code. In prior work [1] by …
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code C, identify for each dimension v, the smallest size of the support of a sub code of C of dimension u, The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work [1] by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei in [2]. This results in a characterization of the GHW hierarchy of binary PRM codes.
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