1,398 research outputs found
Frequency permutation arrays
Motivated by recent interest in permutation arrays, we introduce and
investigate the more general concept of frequency permutation arrays (FPAs). An
FPA of length n=m lambda and distance d is a set T of multipermutations on a
multiset of m symbols, each repeated with frequency lambda, such that the
Hamming distance between any distinct x,y in T is at least d. Such arrays have
potential applications in powerline communication. In this paper, we establish
basic properties of FPAs, and provide direct constructions for FPAs using a
range of combinatorial objects, including polynomials over finite fields,
combinatorial designs, and codes. We also provide recursive constructions, and
give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design
Normalized Leonard pairs and Askey-Wilson relations
Let denote a vector space with finite positive dimension, and let
denote a Leonard pair on . As is known, the linear transformations
satisfy the Askey-Wilson relations A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA
+eI,
B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI, for some scalars
. The scalar sequence is unique if the dimension of is at
least 4. If are scalars and are not zero, then
is a Leonard pair on as well. These affine transformations
can be used to bring the Leonard pair or its Askey-Wilson relations into a
convenient form. This paper presents convenient normalizations of Leonard pairs
by the affine transformations, and exhibits explicit Askey-Wilson relations
satisfied by them.Comment: 22 pages; corrected version, with improved presentation of Section
On unbalanced Boolean functions with best correlation immunity
It is known that the order of correlation immunity of a nonconstant
unbalanced Boolean function in variables cannot exceed ; moreover,
it is if and only if the function corresponds to an equitable
-partition of the -cube with an eigenvalue of the quotient matrix.
The known series of such functions have proportion , , or of
the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean
function attains the correlation-immunity bound and has ratio of the
number of ones and zeros, then is divisible by . In particular, this
proves the nonexistence of equitable partitions for an infinite series of
putative quotient matrices. We also establish that there are exactly
equivalence classes of the equitable partitions of the -cube with quotient
matrix and classes, with . These
parameters correspond to the Boolean functions in variables with
correlation immunity and proportion and , respectively (the case
remains unsolved). This also implies the characterization of the
orthogonal arrays OA and OA.Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse
Counting Steiner triple systems with classical parameters and prescribed rank
By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of
a Steiner triple system on points is at least , and equality
holds only for the classical point-line design in the projective geometry
. It follows from results of Assmus \cite{A} that, given any integer
with , there is a code containing
representatives of all isomorphism classes of STS with 2-rank at most
. Using a mixture of coding theoretic, geometric, design
theoretic and combinatorial arguments, we prove a general formula for the
number of distinct STS with 2-rank at most contained
in this code. This generalizes the only previously known cases, , proved
by Tonchev \cite{T01} in 2001, , proved by V. Zinoviev and D. Zinoviev
\cite{ZZ12} in 2012, and (V. Zinoviev and D. Zinoviev \cite{ZZ13},
\cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and
simplifying the proofs. This enumeration result allows us to prove lower and
upper bounds for the number of isomorphism classes of STS with 2-rank
exactly (or at most) . Finally, using our recent systematic
study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain
analogous results for the ternary case, that is, for STS with 3-rank at
most (or exactly) . We note that this work provides the first
two infinite families of 2-designs for which one has non-trivial lower and
upper bounds for the number of non-isomorphic examples with a prescribed
-rank in almost the entire range of possible ranks.Comment: 27 page
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