433 research outputs found
On set systems with restricted intersections modulo p and p-ary t-designs
We consider bounds on the size of families â± of subsets of a v-set subject to restrictions modulo a prime p on the cardinalities of the pairwise intersections. We improve the known bound when â± is allowed to contain sets of different sizes, but only in a special case. We show that if the bound for uniform families â± holds with equality, then â± is the set of blocks of what we call a p-ary t-design for certain values of t. This motivates us to make a few observations about p-ary t-designs for their own sake
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Universal pâary designs
We investigate -ary -designs which are simultaneously designs for all , which we call universal -ary designs. Null universal designs are well understood due to Gordon James via the representation theory of the symmetric group. We study non-null designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs, up to similarity.Woolf Fisher Trust, Cambridge Commonwealth European and International Trus
Universal pâary designs
We investigate -ary -designs which are simultaneously designs for all , which we call universal -ary designs. Null universal designs are well understood due to Gordon James via the representation theory of the symmetric group. We study non-null designs and determine necessary and sufficient conditions on the coefficients for such a design to exist. This allows us to classify all universal designs, up to similarity.Woolf Fisher Trust, Cambridge Commonwealth European and International Trus
Lower Rate Bounds for Hermitian-Lifted Codes for Odd Prime Characteristic
Locally recoverable codes are error correcting codes with the additional
property that every symbol of any codeword can be recovered from a small set of
other symbols. This property is particularly desirable in cloud storage
applications. A locally recoverable code is said to have availability if
each position has disjoint recovery sets. Hermitian-lifted codes are
locally recoverable codes with high availability first described by Lopez,
Malmskog, Matthews, Pi\~nero-Gonzales, and Wootters. The codes are based on the
well-known Hermitian curve and incorporate the novel technique of lifting to
increase the rate of the code. Lopez et al. lower bounded the rate of the codes
defined over fields with characteristic 2. This paper generalizes their work to
show that the rate of Hermitian-lifted codes is bounded below by a positive
constant depending on when for any odd prime
Complete -term arithmetic progression free sets of small size in vector spaces and other abelian groups
Let denote an Abelian group, written additively. A -term arithmetic
progression of , - for short, is a set of three distinct
elements of of the form , , , for some . A set
is called - free if it does not contain a
-. is called complete - free if it is
- free and the addition of any further element would violate
the - free property. In case is not necessarily
- free but for each there is a
- of consisting of and two elements of , we call
saturating w.r.t. -s. A classical problem in additive
combinatorics is to determine the maximal size of a - free set.
A classical problem in finite geometry and coding theory is to find the minimal
size of a complete cap in affine spaces, where a cap is a point set without a
collinear triplet and in caps and - free sets
are the same objects. In this paper we determine the minimum size of complete
- free sets and - saturating sets for several
families of vector spaces and cyclic groups, up to a small multiplicative
constant factor.Comment: Some typos are corrected, some new references are adde
Beyond developable: computational design and fabrication with auxetic materials
We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications
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