40 research outputs found
On the structure of non-full-rank perfect codes
The Krotov combining construction of perfect 1-error-correcting binary codes
from 2000 and a theorem of Heden saying that every non-full-rank perfect
1-error-correcting binary code can be constructed by this combining
construction is generalized to the -ary case. Simply, every non-full-rank
perfect code is the union of a well-defined family of -components
, where belongs to an "outer" perfect code , and these
components are at distance three from each other. Components from distinct
codes can thus freely be combined to obtain new perfect codes. The Phelps
general product construction of perfect binary code from 1984 is generalized to
obtain -components, and new lower bounds on the number of perfect
1-error-correcting -ary codes are presented.Comment: 8 page
An enumeration of 1-perfect ternary codes
We study codes with parameters of the ternary Hamming
code, i.e., ternary -perfect codes. The rank of
the code is defined to be the dimension of its affine span. We characterize
ternary -perfect codes of rank , count their number, and prove that
all such codes can be obtained from each other by a sequence of two-coordinate
switchings. We enumerate ternary -perfect codes of length obtained by
concatenation from codes of lengths and ; we find that there are
equivalence classes of such codes.
Keywords: perfect codes, ternary codes, concatenation, switching
On decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4
A subset of is called a -fold MDS code if every
line in each of base directions contains exactly elements of . The
adjacency graph of a -fold MDS code is not connected if and only if the
characteristic function of the code is the repetition-free sum of the
characteristic functions of -fold MDS codes of smaller lengths.
In the case , the theory has the following application. The union of two
disjoint MDS codes in is a double-MDS-code. If
the adjacency graph of the double-MDS-code is not connected, then the
double-code can be decomposed into double-MDS-codes of smaller lengths. If the
graph has more than two connected components, then the MDS codes are also
decomposable. The result has an interpretation as a test for reducibility of
-quasigroups of order 4. Keywords: MDS codes, n-quasigroups,
decomposability, reducibility, frequency hypercubes, latin hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to
Discr. Mat
Switching codes and designs
AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Application of advanced on-board processing concepts to future satellite communications systems: Bibliography
Abstracts are presented of a literature survey of reports concerning the application of signal processing concepts. Approximately 300 references are included