1,641 research outputs found
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 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 -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
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