: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

http://www.dtic.mil/dtic/tr/fulltext/u2/a273552.pdf

Discrete Applied Mathematics

Fuzzy/similarity joins have been widely studied in the research community and extensively used in real-world applications. This paper proposes and evaluates several algorithms for finding all pairs of elements from an input set that meet a similarity threshold. The computation model is a single MapReduce job. Because we allow only one MapReduce round, the Reduce function must be designed so a given output pair is produced by only one task, for many algorithms, satisfying this condition is one of the biggest challenges. We break the cost of an algorithm into three components: the execution cost of the mappers, the execution cost of the reducers, and the communication cost from the mappers to reducers. The algorithms are presented first in terms of Hamming distance, but extensions to edit distance and Jaccard distance are shown as well. We find that there are many different approaches to the similarity-join problem using MapReduce, and none dominates the others when both communication and reducer costs are considered. Our cost analyses enable applications to pick the optimal algorithm based on their communication, memory, and cluster requirements.

/pdf/fuzzy-joins-using-mapreduce-2ba7n9p796.pdf

Fuzzy Joins Using MapReduce

A common problem that experimenters face is the choice of fractional factorial designs. Minimum aberration designs are commonly used in practice. There are situations in which other designs meet practical needs better. A catalogue of designs would help experimenters choose the best design. Based on coding theory, new methods are proposed to classify and rank fractional factorial designs efficiently. We have completely enumerated all 27 and 81-run designs, 243-run designs of resolution IV or higher, and 729-run designs of resolution V or higher. A collection of useful fractional factorial designs with 27, 81, 243 and 729 runs is given. This extends the work of Ch93, who gave a collection of fractional factorial designs with 16, 27, 32 and 64 runs.

/pdf/a-catalogue-of-three-level-regular-fractional-factorial-4qz0qyeb7r.pdf

A catalogue of three-level regular fractional factorial designs

Let $\mathcal A$R,q denote a family of covering codes, in
which the covering radius $R$ and the size $q$ of the
underlying Galois field are fixed, while the code length tends
to infinity. The construction of families with small asymptotic
covering densities is a classical problem in the area of
Covering Codes.
   
In this paper, infinite sets of families $\mathcal A$R,q,
where $R$ is fixed but $q$ ranges over an infinite set of prime
powers are considered, and the dependence on $q$ of the
asymptotic covering densities of $\mathcal A$R,q is
investigated. It turns out that for the upper limit 
$\mu$q*(R,$\mathcal A$R,q) of the covering density of
$\mathcal A$R,q, the best possibility is
$\mu$q*(R,$\mathcal A$R,q)=$O(q)$. The main achievement of the
present paper is the construction of optimal infinite
sets of families $\mathcal A$R,q, that is, sets of families
such that relation $\mu$q*(R,$\mathcal A$R,q)=$O(q)$
holds, for any covering radius $R\geq 2$.
   
We first showed that for a given $R$, to obtain optimal
infinite sets of families it is enough to construct $R$
infinite families $\mathcal A$R,q(0),
$\mathcal A$R,q(1), $\ldots$, 
$\mathcal A$R,q(R-1) such that, 
for all $u\geq u$0,
the family $\mathcal A$R,q($\gamma$) contains codes of
codimension $r$u$=Ru + \gamma$ and length 
$f$q($\gamma$)($r$u)
where $f$q($\gamma$)$(r)=O(q$(r-R)/R$)$ and
$u$0 is a constant. Then, we were able to construct the
necessary families $\mathcal A$R,q($\gamma$) for any
covering radius $R\geq 2$, with $q$ ranging over the (infinite)
set of $R$-th powers. A result of independent interest is that
in each of these families $\mathcal A$R,q($\gamma$), the
lower limit of the covering density is bounded from above by a
constant independent of $q$.
   
The key tool in our investigation is the design of new small
saturating sets in projective spaces over finite fields, which
are used as the starting point for the $q$m-concatenating
constructions of covering codes. A new concept of $N$-fold
strong blocking set is introduced. As a result of our
investigation, many new asymptotic and finite upper bounds on
the length function of covering codes and on the smallest sizes
of saturating sets, are also obtained. Updated tables for these
upper bounds are provided. An analysis and a survey of the
known results are presented.

/pdf/linear-nonbinary-covering-codes-and-saturating-sets-in-1gfepjzxps.pdf

Linear nonbinary covering codes and saturating sets in projective spaces

In this paper we present a developed software in the area of Coding Theory. Using it, codes with given properties can be classified. A part of this software can be used also for investigations (isomorphisms, automorphism groups) of other discrete structures-combinatorial designs, Hadamard matrices, bipartite graphs etc.

What is Q-extension?

In this paper we discuss an algorithm for code equivalence. We reduce the equivalence test for linear codes to a test for isomorphism of binary matrices.

/pdf/about-the-code-equivalence-14d7wnn6yt.pdf

About the code equivalence

An efficient algorithm for classification of binary self-dual codes is presented. As an application, a complete classification of the self-dual codes of length 38 is given.

An Algorithm for Classification of Binary Self-Dual Codes

In this paper, we complete the classification of optimal binary linear self-orthogonal codes up to length 25. Optimal self-orthogonal codes are also classified for parameters up to length 40 and dimension 10. The results were obtained via two independent computer searches.

/pdf/classification-of-optimal-binary-self-orthogonal-codes-1jjnlkdiug.pdf

Classification of Optimal Binary Self-Orthogonal Codes

Projective two-weight codes with relatively small parameters are enumerated up to equivalence. Some properties of codes and related strongly-regular graphs are presented.

/pdf/projective-two-weight-codes-with-small-parameters-and-their-3bynci6261.pdf

Iliya Bouyukliev

Papers

What is Q-extension?

About the code equivalence

An Algorithm for Classification of Binary Self-Dual Codes

Classification of Optimal Binary Self-Orthogonal Codes

Projective two-weight codes with small parameters and their corresponding graphs