215 research outputs found
Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays
Constant composition codes have been proposed as suitable coding schemes to
solve the narrow band and impulse noise problems associated with powerline
communication. In particular, a certain class of constant composition codes
called frequency permutation arrays have been suggested as ideal, in some
sense, for these purposes. In this paper we characterise a family of neighbour
transitive codes in Hamming graphs in which frequency permutation arrays play a
central rode. We also classify all the permutation codes generated by groups in
this family
On the number of 1-perfect binary codes: a lower bound
We present a construction of 1-perfect binary codes, which gives a new lower
bound on the number of such codes. We conjecture that this lower bound is
asymptotically tight.Comment: 5pp(Eng)+7pp(Rus) V2: revised V3: + Russian version, + reference
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
Binary Hamming codes and Boolean designs
In this paper we consider a finite-dimensional vector space P over the Galois field GF(2), and the family Bk (respectively, B 17k) of all the k-sets of elements of P (respectively, of P 17=P 16{0}) summing up to zero. We compute the parameters of the 3-design (P,Bk) for any (necessarily even) k, and of the 2-design (P 17,B 17k) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of P, respectively of P 17, that induce permutations of Bk, respectively of B 17k. In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight
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
- …