Two New Four-Error-Correcting Binary Codes

@article{stergrd2005TwoNF,
  title={Two New Four-Error-Correcting Binary Codes},
  author={Patric R. J. {\"O}sterg{\aa}rd},
  journal={Designs, Codes and Cryptography},
  year={2005},
  volume={36},
  pages={327-329}
}
  • P. Östergård
  • Published 1 September 2005
  • Computer Science
  • Designs, Codes and Cryptography
Two four-error-correcting binary codes of length 21 and 22 and of cardinality 64 and 80, respectively, are constructed. The codes consist of a union of cosets of linear codes with dimension 3 and were found by a maximum clique algorithm. 
New Lower Bounds on Error-Correcting Ternary, Quaternary and Quinary Codes
TLDR
19 new lower bounds where q is in 3,4,5, 4,5 are presented, based on codes whose automorphisms are prescribed by transitive permutation groups.
Efficient representation of binary nonlinear codes: constructions and minimum distance computation
TLDR
The complexity of some algorithms to obtain a binary nonlinear code representation based on a union of cosets of a binary linear subcode is analyzed, and some properties and constructions of new codes from given ones in terms of this representation are described.
New lower bounds on q-ary error-correcting codes
TLDR
New lower bounds on and updated tables of Aq (n, d) for q ∈ {3, 4, 5} are presented and groups that act transitively on the (coordinate,value) pairs as well as groups with certain other closely related actions are considered.
A SIMPLE PROOF OF THE IMPROVED JOHNSON BOUND FOR BINARY CODES
TLDR
A simple proof of the improved Johnson bound for A(n, d), the maximum number of codewords in a binary code of length n and minimum distance d, given by Mounits, Etzion and Litsyn is given.
On optimal binary codes with unbalanced coordinates
  • P. Östergård
  • Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2013
TLDR
There are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate, proved by the code attaining A(17,8) = 36 = 36.
RELIABLE COMMUNICATIONS OVER POWER LINES THROUGH CODED MODULATION SCHEMES
TLDR
A new infinite families of optimal equitable symbol weight codes are constructed whose narrowband noise error-correcting capability to code length ratios are bounded away for zero.
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.

References

SHOWING 1-8 OF 8 REFERENCES
New single-error-correcting codes
A matrix construction of nonlinear error-correcting codes is considered. It is shown how this construction and some related theorems can be applied to old codes to get new codes with minimum distance
Two new nonlinear binary codes
TLDR
Two new binary codes are presented: a (25,384,9) code and a (49,393216,13) code that are better than the currently known binary codes with the same length and minimum distance.
Bounds for binary codes of length less than 25
Improved bounds for A(n,d) , the maximum number of codewords in a (linear or nonlinear) binary code of word length n and minimum distance d , and for A(n,d,w) , the maximum number of binary vectors
A table of upper bounds for binary codes
TLDR
Using previous upper bounds on the size of constant-weight binary codes, known methods are reapply to generate a table of bounds on A(n, d) for all n/spl les/28, which extends the range of parameters compared with previously known tables.
Codes from Affine Permutation Groups
TLDR
It is shown that A(n,d) denotes the maximum cardinality of a binary code of length n and minimum Hamming distance d, and the constructed codes are invariant under permutations of some affine (or closely related) permutation group.
Classifying Subspaces of Hamming Spaces
TLDR
This work considers the problem of classifying all [n, k, d]q codes given n,K, d, and q, and classifies all k-dimensional subspaces of the Hamming space with minimum distance at least d given Fnq and a dimension k.
A fast algorithm for the maximum clique problem