Computational Hardness and Explicit Constructions of Error Correcting Codes

Computational Hardness and Explicit Constructions of Error Correcting Codes. Cheraghchi, M., Shokrollahi, A., & Wigderson, A. In Proceedings of the 44th Annual Allerton Conference on Communication, Control, and Computing, pages 1173–1179, 2006.

Paper abstract bibtex

We outline a procedure for using pseudorandom generators to construct binary codes with good properties, assuming the existence of sufficiently hard functions. Specifically, we give a polynomial time algorithm, which for every integers $n$ and $k$, constructs polynomially many linear codes of block length $n$ and dimension $k$, most of which achieving the Gilbert-Varshamov bound. The success of the procedure relies on the assumption that the exponential time class of $\mathrm{E} := \mathrm{DTIME}[2^{O(n)}]$ is not contained in the sub-exponential space class $\mathrm{DSPACE}[2^{o(n)}]$. The methods used in this paper are by now standard within computational complexity theory, and the main contribution of this note is observing that they are relevant to the construction of optimal codes. We attempt to make this note self contained, and describe the relevant results and proofs from the theory of pseudorandomness in some detail.

@INPROCEEDINGS{ref:conf:CSW06,
  author =	 {Mahdi Cheraghchi and Amin Shokrollahi and Avi
                  Wigderson},
  title =	 {Computational Hardness and Explicit Constructions of
                  Error Correcting Codes},
  year =	 2006,
  booktitle =	 {Proceedings of the 44th {Annual Allerton Conference
                  on Communication, Control, and Computing}},
  pages =	 {1173--1179},
  url_Paper =	 {https://infoscience.epfl.ch/record/101078},
  abstract =	 { We outline a procedure for using pseudorandom
                  generators to construct binary codes with good
                  properties, assuming the existence of sufficiently
                  hard functions.  Specifically, we give a polynomial
                  time algorithm, which for every integers $n$ and
                  $k$, constructs polynomially many linear codes of
                  block length $n$ and dimension $k$, most of which
                  achieving the Gilbert-Varshamov bound. The success
                  of the procedure relies on the assumption that the
                  exponential time class of $\mathrm{E} :=
                  \mathrm{DTIME}[2^{O(n)}]$ is not contained in the
                  sub-exponential space class
                  $\mathrm{DSPACE}[2^{o(n)}]$.  The methods used in
                  this paper are by now standard within computational
                  complexity theory, and the main contribution of this
                  note is observing that they are relevant to the
                  construction of optimal codes. We attempt to make
                  this note self contained, and describe the relevant
                  results and proofs from the theory of
                  pseudorandomness in some detail.  }
}

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