179 research outputs found
On the binary codes with parameters of triply-shortened 1-perfect codes
We study properties of binary codes with parameters close to the parameters
of 1-perfect codes. An arbitrary binary code ,
i.e., a code with parameters of a triply-shortened extended Hamming code, is a
cell of an equitable partition of the -cube into six cells. An arbitrary
binary code , i.e., a code with parameters of a
triply-shortened Hamming code, is a cell of an equitable family (but not a
partition) from six cells. As a corollary, the codes and are completely
semiregular; i.e., the weight distribution of such a code depends only on the
minimal and maximal codeword weights and the code parameters. Moreover, if
is self-complementary, then it is completely regular. As an intermediate
result, we prove, in terms of distance distributions, a general criterion for a
partition of the vertices of a graph (from rather general class of graphs,
including the distance-regular graphs) to be equitable. Keywords: 1-perfect
code; triply-shortened 1-perfect code; equitable partition; perfect coloring;
weight distribution; distance distributionComment: 12 page
On Optimal Binary One-Error-Correcting Codes of Lengths and
Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that
triply-shortened and doubly-shortened binary Hamming codes (which have length
and , respectively) are optimal. Properties of such codes are
here studied, determining among other things parameters of certain subcodes. A
utilization of these properties makes a computer-aided classification of the
optimal binary one-error-correcting codes of lengths 12 and 13 possible; there
are 237610 and 117823 such codes, respectively (with 27375 and 17513
inequivalent extensions). This completes the classification of optimal binary
one-error-correcting codes for all lengths up to 15. Some properties of the
classified codes are further investigated. Finally, it is proved that for any
, there are optimal binary one-error-correcting codes of length
and that cannot be lengthened to perfect codes of length
.Comment: Accepted for publication in IEEE Transactions on Information Theory.
Data available at http://www.iki.fi/opottone/code
Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes
The doubly shortened perfect codes of length 13 are classified utilizing the
classification of perfect codes in [P.R.J. \"Osterg{\aa}rd and O. Pottonen, The
perfect binary one-error-correcting codes of length 15: Part I -
Classification, IEEE Trans. Inform. Theory, to appear]; there are 117821 such
(13,512,3) codes. By applying a switching operation to those codes, two more
(13,512,3) codes are obtained, which are then not doubly shortened perfect
codes.Comment: v2: a correction concerning shortened codes of length 1
On -ary shortened--perfect-like codes
We study codes with parameters of -ary shortened Hamming codes, i.e.,
. At first, we prove the fact mentioned in
[A.E.Brouwer et al. Bounds on mixed binary/ternary codes. IEEE Trans. Inf.
Theory 44 (1998) 140-161] that such codes are optimal, generalizing it to a
bound for multifold packings of radius- balls, with a corollary for multiple
coverings. In particular, we show that the punctured Hamming code is an optimal
-fold packing with minimum distance . At second, we show the existence of
-ary codes with parameters of shortened -perfect codes that cannot be
obtained by shortening a -perfect code. Keywords: Hamming graph; multifold
packings; multiple coverings; perfect codes
Local duality for equitable partitions of a Hamming space
AbstractThe Hamming space Qn is the set of binary words of length n. A partition (C1,C2,…,Cr) of Qn with quotient matrix B=[bij]r×r is equitable if for all i and j, any word in the cell Ci has exactly bij neighbors in the cell Cj. In this paper, we provide an explicit formula relating the local spectrum of cells in the face to that in the orthogonal face
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