430 research outputs found
Automorphism groups of Grassmann codes
We use a theorem of Chow (1949) on line-preserving bijections of
Grassmannians to determine the automorphism group of Grassmann codes. Further,
we analyze the automorphisms of the big cell of a Grassmannian and then use it
to settle an open question of Beelen et al. (2010) concerning the permutation
automorphism groups of affine Grassmann codes. Finally, we prove an analogue of
Chow's theorem for the case of Schubert divisors in Grassmannians and then use
it to determine the automorphism group of linear codes associated to such
Schubert divisors. In the course of this work, we also give an alternative
short proof of MacWilliams theorem concerning the equivalence of linear codes
and a characterization of maximal linear subspaces of Schubert divisors in
Grassmannians.Comment: revised versio
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
On the equivalence of linear sets
Let be a linear set of pseudoregulus type in a line in
, or . We provide examples of
-order canonical subgeometries such
that there is a -space with the property that for , is the projection
of from center and there exists no collineation of
such that and .
Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes
Cryptogr. 56:89-104, 2010) states the existence of a collineation between the
projecting configurations (each of them consisting of a center and a
subgeometry), which give rise by means of projections to two linear sets. It
follows from our examples that this condition is not necessary for the
equivalence of two linear sets as stated there. We characterize the linear sets
for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented.
The final version is to appear in Designs, Codes and Cryptograph
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