Simple Codes and Sparse Recovery with Fast Decoding

Simple Codes and Sparse Recovery with Fast Decoding. Cheraghchi, M. & Ribeiro, J. In Proceedings of the IEEE International Symposium on Information Theory (ISIT), pages 156–160, 2019.

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Construction of error-correcting codes achieving a designated minimum distance parameter is a central problem in coding theory. A classical and algebraic family of error-correcting codes studied for this purpose are the BCH codes. In this work, we study a very simple construction of linear codes that achieve a given distance parameter $K$. Moreover, we design a simple, nearly optimal syndrome decoder for the code as well. The running time of the decoder is only logarithmic in the block length of the code, and nearly linear in the distance parameter $K$. This decoder can be applied to exact for-all sparse recovery over any field, improving upon previous results with the same number of measurements. Furthermore, computation of the syndrome from a received word can be done in nearly linear time in the block length. We also demonstrate an application of these techniques in non-adaptive group testing, and construct simple explicit measurement schemes with $O(K^2 łog^2 N)$ tests and $O(K^3 łog^2 N)$ recovery time for identifying up to $K$ defectives in a population of size $N$.

@INPROCEEDINGS{ref:CR19b,
  author =	 {Mahdi Cheraghchi and Jo\~{a}o Ribeiro},
  title =	 {Simple Codes and Sparse Recovery with Fast Decoding},
  year =	 2019,
  doi =		 {10.1109/ISIT.2019.8849702},
  booktitle =	 {Proceedings of the {IEEE International Symposium on
                  Information Theory (ISIT)}},
  pages =	 {156--160},
  url_Link =	 {https://ieeexplore.ieee.org/document/8849702},
  url_Paper =	 {https://arxiv.org/abs/1901.02852},
  abstract =	 {Construction of error-correcting codes achieving a
                  designated minimum distance parameter is a central
                  problem in coding theory. A classical and algebraic
                  family of error-correcting codes studied for this
                  purpose are the BCH codes.  In this work, we study a
                  very simple construction of linear codes that
                  achieve a given distance parameter $K$. Moreover, we
                  design a simple, nearly optimal syndrome decoder for
                  the code as well. The running time of the decoder is
                  only logarithmic in the block length of the code,
                  and nearly linear in the distance parameter $K$.
                  This decoder can be applied to exact for-all sparse
                  recovery over any field, improving upon previous
                  results with the same number of measurements.
                  Furthermore, computation of the syndrome from a
                  received word can be done in nearly linear time in
                  the block length.  We also demonstrate an
                  application of these techniques in non-adaptive
                  group testing, and construct simple explicit
                  measurement schemes with $O(K^2 \log^2 N)$ tests and
                  $O(K^3 \log^2 N)$ recovery time for identifying up
                  to $K$ defectives in a population of size $N$.  }
}

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