Weight Spectrum of Quasi-Perfect Binary Codes with Distance 4

We consider the weight spectrum of a class of quasi-perfect binary linear codes with code distance 4. For example, extended Hamming code and Panchenko code are the known members of this class. Also, it is known that in many cases Panchenko code has the minimal number of weight 4 codewords. We give exact recursive formulas for the weight spectrum of quasi-perfect codes and their dual codes. As an example of application of the weight spectrum we derive a lower estimate for the conditional probability of correction of erasure patterns of high weights (equal to or greater than code distance).Comment: 5 pages, 11 references, 2 tables; some explanations and detail are adde

New sizes of complete arcs in PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853<= q<= 2879 and q=3511,4096, 4523,5003,5347,5641,5843,6011. For q<= 2377 and q=2401,2417,2437, the relation t_{2}(2,q)<4.5\sqrt{q} holds. The bounds are obtained by finding of new small complete arcs with the help of computer search using randomized greedy algorithms. Also new sizes of complete arcs are presented.Comment: 10 page

On upper bounds on the smallest size of a saturating set in a projective plane

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\Pi _{q}$ is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant $c\ge 1$ a random point set of size $k$ in $\Pi _{q}$ with $2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q$ is a saturating set with probability greater than $1-1/(q+1)^{2c^{2}-2}.$ Our probabilistic approach is also applied to multiple saturating sets. A point set $S\subset \Pi_{q}$ is $(1,\mu)$-saturating if for every point $Q$ of $\Pi _{q}\setminus S$ the number of secants of $S$ through $Q$ is at least $\mu$, counted with multiplicity. The multiplicity of a secant $\ell$ is computed as ${\binom{{\#(\ell \,\cap S)}}{{2}}}.$ The following upper bound on the smallest size $s_{\mu }(2,q)$ of a $(1,\mu)$-saturating set in $\Pi_{q}$ is proved: \begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a $(1,\mu)$-saturating set) in the projective space $PG(N,q)$ are obtained. All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and some references are adde

On sizes of complete arcs in PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853 <= q <= 4561 and q\in T1\cup T2 where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457, 461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631, 641,661,673,677,683,691, 709}, T2={4597,4703,4723,4733,4789,4799,4813,4831,5003,5347,5641,5843,6011,8192}. From these new bounds it follows that for q <= 2593 and q=2693,2753, the relation t_{2}(2,q) < 4.5\sqrt{q} holds. Also, for q <= 4561 we have t_{2}(2,q) < 4.75\sqrt{q}. It is showed that for 23 <= q <= 4561 and q\in T2\cup {2^{14},2^{15},2^{18}}, the inequality t_{2}(2,q) < \sqrt{q}ln^{0.75}q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q >= 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} <= k <= k_{max} where k_{min} is of order q/3 or q/4 while k_{max} has order q/2. The completeness of the arcs obtained by the new constructions is proved for q <= 1367 and 2003 <= q <= 2063. There is reason to suppose that the arcs are complete for all q > 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.Comment: 27 pages, 4 figures, 5 table