Self-orthogonal Latin Squares Research Articles

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR homogeneous.In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive.We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property.Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

A novel image encryption algorithm based on self-orthogonal Latin squares

This paper proposes a novel image encryption algorithm based on self-orthogonal Latin squares. A self-orthogonal Latin square can generate a 2D map and some 1D maps which can be used for permutations of image. It can also provide a pseudo-random sequence which can be used for substitutions of image. Simulation results show that the proposed algorithm has not only a desirable level of security, but also high efficiency, so the algorithm is suitable for practical application.

25 new [formula omitted]-self-orthogonal Latin squares

Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If one of the two squares is the transpose of the other, we say that the square is r-self-orthogonal, denoted by r-SOLS(n). It has been proved by Xu and Chang that the necessary and sufficient condition for the existence of an r-SOLS(n) is n≤r≤n2 and r∉{n+1,n2−1} with 26 genuine exceptions and 26 possible exceptions. In this paper, we provide 25 new Latin squares to reduce the possible exceptions from 26 to one, i.e., (n,r)=(14,142−3). We also provide an idempotent incomplete self-orthogonal Latin square (ISOLS) of order 26 with a hole of size 8.

Existence of 2 SOLS and 2 ISOLS

In this paper, we investigate the existence of two self-orthogonal Latin squares of order v (2- SOLS ( v ) ). It is found that 2- SOLS ( v ) exist for all v ≥ 7 except possibly for v ∈ { 10 , 12 , 14 , 18 , 21 , 22 , 24 , 30 , 34 } . A number of sets of 2 incomplete SOLS (2 ISOLS) (and hence also 4 incomplete mutually orthogonal Latin squares, briefly 4 IMOLS) are also given.

Mixed group divisible designs with three groups and block size 4

A group divisible design (GDD) with three groups and block size 4 is called even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups are always even, always odd, or always mixed. It has been shown that the necessary conditions for the existence of GDDs of these three types are also sufficient except possibly for the minimal case of mixed designs for group size 5 t ( t > 1 ). In this paper, we complete the undetermined families of mixed GDDs using two constructions based on idempotent self-orthogonal Latin squares and skew Room squares.

A graph-theoretic proof of the non-existence of self-orthogonal Latin squares of order 6

The non-existence of a pair of mutually orthogonal Latin squares of order six is a well-known result in the theory of combinatorial designs. It was conjectured by Euler in 1782 and was first proved by Tarry in 1900 by means of an exhaustive enumeration of equivalence classes of Latin squares of order six. Various further proofs have since been given, but these proofs generally require extensive prior subject knowledge in order to follow them, or are ‘blind’ proofs in the sense that most of the work is done by computer or by exhaustive enumeration. In this paper we present a graph-theoretic proof of a somewhat weaker result, namely the non-existence of self-orthogonal Latin squares of order six, by introducing the concept of a self-orthogonal Latin square graph. The advantage of this proof is that it is easily verifiable and accessible to discrete mathematicians not intimately familiar with the theory of combinatorial designs. The proof also does not require any significant prior knowledge of graph theory.

Existence of HSOLSSOMs of type [formula omitted

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2 n u 1 ( HSOLSSOM ( 2 n u 1 ) ). For u ⩾ 2 , necessary conditions for existence of such an HSOLSSOM are that u must be even and n ⩾ 3 u / 2 + 1 . Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n ⩽ 9 and n ⩾ 3 u / 2 + 1 or (2) n ⩾ 263 and n ⩾ 2 ( u - 2 ) . In this paper we show that in (1) the condition n ⩽ 9 can be extended to n ⩽ 30 and that in (2), the condition n ⩾ 263 can be improved to n ⩾ 4 , except possibly for 19 pairs ( n , u ) , the largest of which is ( 53 , 28 ) .

Atomic Latin squares of order eleven

AbstractA Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are pan‐Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1‐factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan‐Hamiltonian Latin square of order n describes a perfect 1‐factorization of Kn,n, and vice versa. Perfect 1‐factorizations of Kn,n can be constructed from a perfect 1‐factorization of Kn+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn‐square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self‐orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self‐orthogonal Latin squares in the same main class as a given Latin square. © 2003 Wiley Periodicals, Inc.

Existence of Steiner seven-cycle systems

A Steiner k-cycle system of order v is a pair (X,C), where C is a collection of k-cycles of K v based on a v-set X such that for any integer r, 1⩽r⩽k/2 , and for any two distinct vertices x and y of X there exists in C a unique k-cycle along which the distance between x and y is r. Steiner k-cycle systems are useful in constructing authentication perpendicular arrays and authentication and secrecy codes. In this paper, we show that the necessary condition for the existence of Steiner seven-cycle systems, v≡1 or 7 ( mod 14) , is also sufficient if v>861. We also show that there are at most 21 unknown orders below this bound. The result is mainly based on generalized constructions for two holey self-orthogonal Latin squares with symmetric orthogonal mates (2 HSOLSSOM) and some direct constructions. As an application, we shall update the known result on the existence of perfect Mendelsohn designs with block size 7.

Constructing self-conjugate self-orthogonal diagonal latin squares

In this paper, we give some constructions of self-conjugate self-orthogonal diagonal Latin squares (SCSODLS). As an application of such constructions we disproof the conjecture about SCSODLS and show that there exist SCSODLS of orderV, whenevervэ1 (mod 12), with the possible exception ofve{13, 85, 133}.

Stellar permutations of the two-element subsets of a finite set

Let X be a finite set and denote by X(2) the set of 2-element subsets of X. A permutation ϕ of X(2) is called stellar if, for each x in X, the image under ϕ of the star St(x) = {{x, y}: x ≠ y ∈ X} is a 2-regular graph spanning X − {x}. Several constructions of stellar permutations are given; in particular, there is a natural direct construction using self-orthogonal Latin squares, and a simple recursive construction using linear spaces having all line sizes at least four. Apart from some intrinsic interest, stellar permutations arise in the construction of certain designs. For example, applying such a map ϕ to each of the stars St(x) yields a double covering of the complete graph on X by near 2-factors. We also study stellar groups, that is groups {ϕ1, …, ϕs} of permutations of X(2) such that each ϕi is stellar (or the identity map). It is elementary to prove that s ≤ for any stellar group; when equality holds, one can construct an associated elation semibiplane. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 381–387, 1998

On Orthogonal Double Covers of Graphs

A transformation which allows us to obtain an orthogonal double cover of a graph G from any permutation of the edge set of G is described. This transformation is used together with existence results for self-orthogonal latin squares, to give a simple proof of a conjecture of Chung and West.

Stellar permutations of the two‐element subsets of a finite set

Let X be a finite set and denote by X(2) the set of 2-element subsets of X. A permutation ϕ of X(2) is called stellar if, for each x in X, the image under ϕ of the star St(x) = {{x, y}: x ≠ y ∈ X} is a 2-regular graph spanning X − {x}. Several constructions of stellar permutations are given; in particular, there is a natural direct construction using self-orthogonal Latin squares, and a simple recursive construction using linear spaces having all line sizes at least four. Apart from some intrinsic interest, stellar permutations arise in the construction of certain designs. For example, applying such a map ϕ to each of the stars St(x) yields a double covering of the complete graph on X by near 2-factors. We also study stellar groups, that is groups {ϕ1, …, ϕs} of permutations of X(2) such that each ϕi is stellar (or the identity map). It is elementary to prove that s ≤ for any stellar group; when equality holds, one can construct an associated elation semibiplane. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 381–387, 1998

Holey self-orthogonal Latin squares with symmetric orthogonal mates

We improve the existence results for holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOMs) and show that the necessary conditions for the existence of a HSOLSSOM of typehn are also sufficient with at most 28 pairs (h, n) of possible exceptions.

The spectrum of HSOLSSOM( hn) where h is even

In this paper, we describe a generalized product construction and a construction using Steiner pentagon systems to obtain holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOM). We investigate the existence of HSOLSSOM( h n ) for even h. We first improve the known result for h = 2 and show that a HSOLSSOM(2 n ) exists for all n ⩾ 5 except possibly for n ∈ E = {8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 32}. As a consequence, we then establish that for h  2 (mod 4) and h ⩾ 10 a HSOLSSOM( h n ) exists for all n ⩾ 5 except possibly for n ϵ E. More conclusively, we show that for h  0 (mod 4), a HSOLSSOM( h n ) exists if and only if n ⩾ 5. We are also able to apply our results to construct a unipotent SOLSSOM(62), the existence of which was previously unknown.

Incomplete self-orthogonal latin squares ISOLS(6 m + 6, 2 m) exist for all m

An incomplete self-orthogonal latin square of order v with an empty subarray of order n, an ISOLS( v, n) can exist only if v ⩾ 3 n + 1. We show that an ISOLS(6 m + 6, 2 m) exists for all values of m and thus only the existence of an ISOLS(6 m + 6, 2 m), m ⩾ 2, remains in doubt.

Incomplete self-orthogonal latin squares

AbstractWe show that for all n ≥ 3k + 1, n ≠ 6, there exists an incomplete self-orthogonal latin square of order n with an empty order k subarray, called an ISOLS(n;k), except perhaps when (n;k) ∈ {(6m + i;2m):i = 2, 6}.

On the existence of MOLS with equal-sized holes

We consider a pair of MOLS (mutually orthogonal Latin squares) having holes, corresponding to missing sub-MOLS, which are disjoint and spanning It is shown that a pair of MOLS withn holes of sizeh exist forh ≥ 2 if and only ifn ≥ 4 For SOLS (self-orthogonal Latin squares) with holes, we have the same result, with two possible exceptions SOLS with 7 or 13 holes of size 6

Resolvable bibd and sols

The ideas of a base factorization and a resolvable grid system are introduced, and a construction of a resolvable balanced incomplete block design (BIBD) from these structures is given. Resolvable grid systems can be constructed from mutually orthogonal self-orthogonal latin squares (SOLS) with symmetric mate. Together these results prove, as a special case, that: if k−1 is an odd prime power and there exist 1 2 (k−2) mutually orthogonal SOLS of order n, with symmetric mate, then there exists a resolvable BIBD with block size k on v = kn points of index λ, where λ = k−1 if k = 0 (mod 4) and λ = 2( k−1) if k = 2 (mod 4). The technique is illustrated for k = 4, λ = 3 and k = 6, λ = 10, in which cases v = 0 (mod k) is shown to be a necessary and sufficient condition (NASC) for the existence of a resolvable BIBD on v points. The pair ( k, λ) = (6,10) thus becomes only the fifth pair for which NASC are known, the other pairs being (3,1), (4,1), (3,2), and (4,3).

Self-orthogonal latin squares of all orders 𝑛≠2,3,6

Self-orthogonal latin squares of all orders 𝑛≠2,3,6