5,190 research outputs found
Design Lines
The two basic equations satisfied by the parameters of a block design define
a three-dimensional affine variety in . A point
of that is not in some sense trivial lies on four lines lying in
. These lines provide a degree of organization for certain general
classes of designs, and the paper is devoted to exploring properties of the
lines. Several examples of families of designs that seem naturally to follow
the lines are presented.Comment: 16 page
Hadamard Equiangular Tight Frames
An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix. These include harmonic ETFs, which are obtained by extracting the rows of a character table that correspond to a difference set in the underlying finite abelian group. In this paper, we give some new results about flat ETFs. One of these results gives an explicit Naimark complement for all Steiner ETFs, which in turn implies that all Kirkman ETFs are possibly-complex Hadamard ETFs. This in particular produces a new infinite family of real flat ETFs. Another result establishes an equivalence between real flat ETFs and certain types of quasi-symmetric designs, resulting in a new infinite family of such designs
Higher-order CIS codes
We introduce {\bf complementary information set codes} of higher-order. A
binary linear code of length and dimension is called a complementary
information set code of order (-CIS code for short) if it has
pairwise disjoint information sets. The duals of such codes permit to reduce
the cost of masking cryptographic algorithms against side-channel attacks. As
in the case of codes for error correction, given the length and the dimension
of a -CIS code, we look for the highest possible minimum distance. In this
paper, this new class of codes is investigated. The existence of good long CIS
codes of order is derived by a counting argument. General constructions
based on cyclic and quasi-cyclic codes and on the building up construction are
given. A formula similar to a mass formula is given. A classification of 3-CIS
codes of length is given. Nonlinear codes better than linear codes are
derived by taking binary images of -codes. A general algorithm based on
Edmonds' basis packing algorithm from matroid theory is developed with the
following property: given a binary linear code of rate it either provides
disjoint information sets or proves that the code is not -CIS. Using
this algorithm, all optimal or best known codes where and are shown to be -CIS for all
such and , except for with and with .Comment: 13 pages; 1 figur
Some irreducible 2-modular codes invariant under the symplectic group S6(2)
We examine all non-trivial binary codes and designs obtained from the 2-modular primitive permutation representations of degrees up to 135 of the simple projective special symplectic group S6(2). The submodule lattice of the permutation modules, together with a comprehensive description of each code including the weight enumerator, the automorphism group, and the action of S6(2) is given. By considering the structures of the stabilizers of several codewords we attempt to gain an insight into the nature of some classes of codewords in particular those of minimum weight
Bent Vectorial Functions, Codes and Designs
Bent functions, or equivalently, Hadamard difference sets in the elementary
Abelian group (\gf(2^{2m}), +), have been employed to construct symmetric and
quasi-symmetric designs having the symmetric difference property. The main
objective of this paper is to use bent vectorial functions for a construction
of a two-parameter family of binary linear codes that do not satisfy the
conditions of the Assmus-Mattson theorem, but nevertheless hold -designs. A
new coding-theoretic characterization of bent vectorial functions is presented
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