6 research outputs found
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
Asymptotics for the number of n-quasigroups of order 4
The asymptotic form of the number of n-quasigroups of order 4 is . Keywords: n-quasigroups, MDS codes, decomposability,
reducibility.Comment: 15 p., 3 fi
On reducibility of n-ary quasigroups
An -ary operation is called an -ary quasigroup of order
if in the equation knowledge of any elements
of , ..., uniquely specifies the remaining one. is permutably
reducible if
where
and are -ary and -ary quasigroups, is a permutation, and
. An -ary quasigroup is called a retract of if it can be
obtained from or one of its inverses by fixing arguments. We prove
that if the maximum arity of a permutably irreducible retract of an -ary
quasigroup belongs to , then is permutably reducible.
Keywords: n-ary quasigroups, retracts, reducibility, distance 2 MDS codes,
latin hypercubesComment: 13 pages; presented at ACCT'2004 v2: revised; bibliography updated; 2
appendixe