Semidefinite Code Bounds Based on Quadruple Distances

@article{Gijswijt2012SemidefiniteCB,
  title={Semidefinite Code Bounds Based on Quadruple Distances},
  author={Dion Gijswijt and Hans D. Mittelmann and Alexander Schrijver},
  journal={IEEE Transactions on Information Theory},
  year={2012},
  volume={58},
  pages={2697-2705}
}
Let <i>A</i>(<i>n</i>,<i>d</i>) be the maximum number of 0, 1 words of length <i>n</i> , any two having Hamming distance at least <i>d</i>. It is proved that <i>A</i>(20,8)=256, which implies that the quadruply shortened Golay code is optimal. Moreover, it is shown that <i>A</i>(18,6) ≤ 673, <i>A</i>(19,6) ≤ 1237, <i>A</i>(20,6) ≤ 2279, <i>A</i>(23,6) ≤ 13674, <i>A</i>(19,8) ≤ 135, <i>A</i>(25,8) ≤ 5421, <i>A</i>(26,8) ≤ 9275, <i>A</i>(27,8) ≤ 17099, <i>A</i>(21,10) ≤ 47, <i>A</i>(22,10) ≤ 84… 

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References

SHOWING 1-10 OF 31 REFERENCES

Strengthened semidefinite programming bounds for codes

  • M. Laurent
  • Computer Science, Mathematics
    Math. Program.
  • 2007
TLDR
Two strengthenings of Schrijver’s bound can be computed via a semidefinite program of size O(n3), a result which uses the explicit block-diagonalization of the Terwilliger algebra.

New code upper bounds from the Terwilliger algebra and semidefinite programming

  • A. Schrijver
  • Computer Science, Mathematics
    IEEE Transactions on Information Theory
  • 2005
We give a new upper bound on the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. It is based on block-diagonalizing the Terwilliger algebra of the Hamming cube.

On the Shannon capacity of a graph

  • L. Lovász
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
TLDR
It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.

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.

The Lovasz bound and some generalizations

TLDR
An extremely powerful and general technique phased in terms of graph theory, for studying combinatorial packing problems is presented, and Delsarte's linear programming bound for cliques in association schemes appears as a special case of the Lovasz bound.

An Explicit Equivalent Positive Semidefinite Program for Nonlinear 0-1 Programs

TLDR
This work considers the general nonlinear optimization problem in 0-1 variables and provides an explicit equivalent positive semidefinite program in 2n-1 variable and its optimal values are identical.

Variational calculation of second-order reduced density matrices by strong N-representability conditions and an accurate semidefinite programming solver.

TLDR
This work explores the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions and provides physically meaningful results for the Hubbard model in the high correlation limit.

A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems

TLDR
It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method, and a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions.

A hierarchy of relaxation between the continuous and convex hull representations

In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into