Energy-Minimizing Error-Correcting Codes

@article{Cohn2014EnergyMinimizingEC,
  title={Energy-Minimizing Error-Correcting Codes},
  author={Henry Cohn and Yufei Zhao},
  journal={IEEE Transactions on Information Theory},
  year={2014},
  volume={60},
  pages={7442-7450}
}
  • Henry Cohn, Yufei Zhao
  • Published 9 December 2012
  • Computer Science
  • IEEE Transactions on Information Theory
We study a discrete model of repelling particles, and we show using linear programming bounds that many familiar families of error-correcting codes minimize a broad class of potential energies when compared with all other codes of the same size and block length. Examples of these universally optimal codes include Hamming, Golay, and Reed-Solomon codes, among many others, and this helps to explain their robustness as the channel model varies. Universal optimality of these codes is equivalent to… 
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References

SHOWING 1-10 OF 31 REFERENCES
A survey of perfect codes
  • O. Heden
  • Computer Science
    Adv. Math. Commun.
  • 2008
TLDR
This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes.
Bounds for unrestricted codes, by linear programming
TLDR
The present paper starts from the observation that the distance distribution of an unrestricted code (over an unrestricted alphabet) yields nonnegative numbers when substituted to the right-hand member of (1) and gives an upper bound for the number of words in codes having a designed set of distances.
Theory of Error-correcting Codes
TLDR
This course expounds the principles of coded modulations for the Gaussian channel and, if time permits, for Rician and Rayleigh fading channels (fully interleaved), and reminds students of the basics of the theory of linear codes for conventional memoryless ergodic channels.
Spin-glass models as error-correcting codes
TLDR
It is shown that theoretical models of magnetically disordered materials (spin glasses) provide a new class of error-correction codes that can, under certain circumstances, constitute the first known codes to saturate Shannon's well-known cost-performance bounds.
Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces
Universal bounds for the cardinality of codes in the Hamming space F/sub r//sup n/ with a given minimum distance d and/or dual distance d' are stated. A self-contained proof of optimality of these…
Linear programming-based optimization of the distance spectrum of linear block codes
TLDR
The results suggest that the best distance spectrum is not a function of signal-to-noise ratio (SNR) above the cutoff rate SNR and also suggest the existence of several unknown, good codes.
Binomial moments of the distance distribution: bounds and applications
  • A. Ashikhmin, A. Barg
  • Computer Science, Mathematics
    Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252)
  • 1998
TLDR
New lower bounds are obtained on the probability of undetected error of binary codes on the binary symmetric channel (BSC) which improve previously known results and derive new asymptotic upper bounds on the exponent of undetection error.
Association Schemes and Coding Theory
TLDR
This paper contains a survey of association scheme theory (with its algebraic and analytical aspects) and of its applications to coding theory (in a wide sense); some "universal bounds" are derived for codes and designs in polynomial type association schemes.
Three-point bounds for energy minimization
TLDR
It is proved that the seven lines connecting opposite vertices of a cube and of its dual octahedron are universally optimal, which is the only known universal optimum that is not distance regular, and the last remaining universal optimum in RP^2.
New Conjectural Lower Bounds on the Optimal Density of Sphere Packings
TLDR
An optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies to obtain upper bounds on the density, and hence has implications for linear programming bounds, and proves that the density estimate can never exceed the Cohn– Elkies upper bound.
...
...