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A covering array CA(N; t; k; v) is an N x k array on v symbols such that every N x t subarray contains as a row each t-tuple over the v symbols at least once. The minimum N for which a CA(N; t; k; v) exists is called the covering array number of t, k, and v, and it is denoted by CAN(t; k; v). In this work we prove new CANs using an optimized procedure., Comment: UNSUBMITTED

Let [math] . For each [math] , Thas and Van Maldeghem constructed a [math] -dimensional dual hyperoval in [math] with [math] , called the Veronesean dual hyperoval. A quotient of the Veronesean dual hyperoval with ambient space [math] , denoted [math] , is constructed by Taniguchi, using a generator [math] of the Galois group Gal [math] . ¶ In this note, using the above generator [math] for [math] and a [math] -dimensional vector subspace [math] of [math] over [math] , we construct a quotient [math] of the Veronesean dual hyperoval in [math] in case [math] is even. Moreover, we prove the following: for generators [math] and [math] of the Galois group Gal [math] , ¶ S σ above (for q = 2 ) is not isomorphic to S τ , H , S σ , H is isomorphic to S σ , H ′ for any d -dimensional vector subspaces H and H ′ of GF ( 2 d + 1 ) , and S σ , H is isomorphic to S τ , H if and only if σ = τ or σ = τ − 1 . ¶ Hence, we construct many new non-isomorphic quotients of the Veronesean dual hyperoval in [math] .

We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be obtained from it by repeated shortening. Further, we generalize the results to partitions: every partition of the Hamming space into $1$-error-correcting codes can be embedded in a partition of a space of some larger dimension into $1$-perfect codes. For the partitions, the embedding length is close to the theoretical bound for the general case and optimal for the binary case. Keywords: error-correcting code, $1$-perfect code, $1$-perfect partition, embedding, Comment: 7 pp

In this paper the 3-way intersection problem for $S(2,4,v)$ designs is investigated. Let $b_{v}=\frac {v(v-1)}{12}$ and $I_{3}[v]=\{0,1,...,b_{v}\}\setminus\{b_{v}-7,b_{v}-6,b_{v}-5,b_{v}-4,b_{v}-3,b_{v}-2,b_{v}-1\}$. Let $J_{3}[v]=\{k|$ there exist three $S(2,4,v)$ designs with $k$ same common blocks$\}$. We show that $J_{3}[v]\subseteq I_{3}[v]$ for any positive integer $v\equiv1, 4\ (\rm mod \ 12)$ and $J_{3}[v]=I_{3}[v]$, for $ v\geq49$ and $v=13 $. We find $J_{3}[16]$ completely. Also we determine some values of $J_{3}[v]$ for $\ v=25,28,37$ and 40., accepted in Utilitas mathematics

We show that, for all odd natural numbers $N$, the $N$-torsion points on an elliptic curve may be placed in an $N \times N$ grid such that the sum of each column and each row is the point at infinity.

We claim that $M$(atroid) theory may provide a mathematical framework for an underlying description of $M$-theory. Duality is the key symmetry which motivates our proposal. The definition of an oriented matroid in terms of the Farkas property plays a central role in our formalism. We outline how this definition may be carried over $M$-theory. As a consequence of our analysis we find a new type of action for extended systems which combines dually the $p$-brane and its dual $p^{\perp}$-brane., Comment: 12 pages, Latex; some improvements introduced, typos corrected