Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$ . A $q$ -analog Steiner system (also known as - ), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$ , is set ${\mathcal{S}}$ $k$ -dimensional subspaces such that each $t$ subspace contained in exactly one element Presently, -Steiner systems are only for $t\,=\,1\!$ and trivial cases $t\,=\,k$ $k\,=\,n$ In this paper, first nontrivial with $t\,\geqslant \,2$ constructed. Specifically, several...

Given a graph, in the maximum clique problem one wants to find largest number of vertices, any two which are adjacent. In maximum-weight problem, vertices have positive, integer weights, and with weight. A recent algorithm for is here used as basis developing an weighted case. Computational experiments random graphs show that this new faster than earlier algorithms many cases. set obtained from constructing good constant weight error-correcting codes proposed test cases algorithms; fact,...

Let $\F_q^n$ be a vector space of dimension $n$ over the finite field $\F_q$. A $q$-analog Steiner system (briefly, $q$-Steiner system), denoted $S_q[t,k,n]$, is set $S$ $k$-dimensional subspaces such that each $t$-dimensional subspace contained in exactly one element $S$. Presently, systems are known only for $t=1$, and trivial cases $t = k$ $k n$. Invthis paper, first nontrivial with >= 2$ constructed. Specifically, several nonisomorphic $S_2[2,3,13]$ found by requiring their...

Upper and lower bounds are presented for the maximal possible size of mixed binary/ternary error-correcting codes. A table up to length 13 is included. The upper obtained by applying linear programming bound product two association schemes. arise from a number different constructions.

Using an orderly algorithm, the Steiner triple systems of order<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="19"><mml:semantics><mml:mn>19</mml:mn><mml:annotation encoding="application/x-tex">19</mml:annotation></mml:semantics></mml:math></inline-formula>are classified; there are<inline-formula alttext="11 comma 084 874 829"><mml:semantics><mml:mrow><mml:mn>11</mml:mn><mml:mo>,</mml:mo><mml:mspace width="negativethinmathspace"...

Constructive and nonconstructive techniques are employed to enumerate Latin squares related objects. It is established that there (i) $2036029552582883134196099$ main classes of order $11$$;$ (ii) $6108088657705958932053657$ isomorphism one-factorizations $K_{11,11}$$;$ (iii) $12216177315369229261482540$ isotopy (iv) $1478157455158044452849321016$ loops (v) $19464657391668924966791023043937578299025$ quasigroups $11$. The enumeration constructive for the $1151666641$ with an autoparatopy...

Let $\gamma_{m,n}$ denote the size of a minimum dominating set in $m \times n$ grid graph. For square graph, exact values for $\gamma_{n,n}$ have earlier been published $n \leq 19$. By using dynamic programming algorithm, $m,n 29$ are here obtained. Minimum sets graphs up to $29 depicted.

The problem of determining the maximum size a ternary code is considered, under restriction that each symbol should appear given number times in codeword. Upper and lower bounds on such codes Hamming metric are discussed, where follow from constructions good codes. Some results obtained by explicitly finding computer search. A table exact values best known for length at most 10 presented.

A complete classification of the perfect binary one-error-correcting codes length 15 as well their extensions 16 is presented. There are 5983 such inequivalent and 2165 extended codes. Efficient generation these relies on recent Steiner quadruple systems order 16. Utilizing a result Blackmore, optimal 14 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(15,</i> xmlns:xlink="http://www.w3.org/1999/xlink">1024,</i>...

Properties of the 11$\,$084$\,$874$\,$829 Steiner triple systems order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); uniform two STS(19) with almost parallel classes; all have chromatic number 3; index 10, except for 4$\,$075 designs 11 and 12; 3-resolvable; 3-existentially closed STS(19).

LCLs or locally checkable labelling problems (e.g. maximal independent set, matching, and vertex colouring) in the LOCAL model of computation are very well-understood cycles (toroidal 1-dimensional grids): every problem has a complexity O(1), Θ(log* n), Θ(n), design optimal algorithms can be fully automated. This work develops theory LCL for toroidal 2-dimensional grids. The classes same as case: Θ(n). However, given an it is undecidable whether its n) Θ(n)

Abstract A graph is hypohamiltonian if it not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, smallest known planar had 42 vertices, result due to Araya and Wiener. That here improved upon by 25 graphs order 40, which are found through computer‐aided generation certain families with girth 4 fixed number 4‐faces. It further shown that exist for all orders greater than or equal 42. If cycles replaced paths throughout definition graphs, we get...

This chapter outlines some of the main computational methods that have been successful in solving existence and enumeration problems combinatorial design theory. Issues related to validation search results are also discussed.

A complete classification of the perfect binary one-error-correcting codes length 15 as well their extensions 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The 15: Part I--Classification," IEEE Trans. Inform. Theory vol. 55, pp. 4657--4660, 2009]. In current accompanying work, classified are studied great detail, main properties tabulated. The results include fact that 33 80 Steiner triple systems order occur such codes. Further understanding is gained on...

A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ a maximum separable (MDS) In this work, some MDS codes over small alphabets are classified. It shown that every $(k+d-1,q^k,d)_q$ with $k\geq 3$, $d \geq $q \in \{5,7\}$ equivalent to linear the same parameters. This implies $(6,5^4,3)_5$ $(n,7^{n-2},3)_7$ for $n\in\{6,7,8\}$ unique. The classification one-error-correcting $8$-ary also finished; there $14$, $8$, $4$, $4$...

Many of the fundamental coding problems can be represented as graph problems. These are often intrinsically difficult and unsolved even if code length is relatively small. With motivation to improve lower bounds on sizes constant weight codes asymmetric codes, we suggest a few greedy algorithms other heuristic methods, which result in new, record-breaking codes. Some heuristics used based tabu search evolutionary algorithms. Tables new presented.

A binary code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ⊆ F <sub xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> xmlns:xlink="http://www.w3.org/1999/xlink">n</i> with minimum distance at least xmlns:xlink="http://www.w3.org/1999/xlink">d</i> and codewords of Hamming weight xmlns:xlink="http://www.w3.org/1999/xlink">w</i> is called an xmlns:xlink="http://www.w3.org/1999/xlink">(n</i> , ) constant code. The maximum size denoted by...