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In this paper a family of novel spreading code called Latin square code (LSC) is proposed for spectral amplitude coding-optical code division multiple access (SAC-OCDMA) system. The main feature of the proposed code is the zero cross-correlation which eliminates both multiple access interference (MAI) and phase induced intensity noise (PIIN). The code construction can be easily accomplished using Latin square matrix (LSM) for any weight and number of users. The simplicity in the construction code has made it a compelling candidate for future OCDMA applications. SAC-OCDMA system employing direct decoding is mathematically analyzed and then numerically simulated using Matlab and OptiSystem respectively. The results show the improvement given by the LSC code to the SAC-OCDMA system compared to the other codes such as: ZCC (zero cross-correlation) and MMS (modified multiservice) by allowing high cardinality and improving BER. Furthermore, the transmission quality, so that the BER does not exceed the value of 10-9, is ensured by the LSC code with a lower effective source power of -14.5 dBm, a bit rate of 4.6 Gb/s and a cardinality exceeding ZCC and MMS by 1.7 and 1.22 times respectively. The simulation results validate the mathematical analysis and show that the system makes it possible to increase the transmission distance without affecting QoS (quality of service). [ABSTRACT FROM AUTHOR]

Abstract We provide an upper bound on the number of n 2 × n 2 Sudoku squares, and explain intuitively why there is reason to believe that the bound is tight up to a multiplicative factor of a much smaller order of magnitude. A similar bound is established for Sudoku squares with rectangular regions. [ABSTRACT FROM AUTHOR]

We say that a diagonal in an array is λ -balanced if each entry occurs λ times. Let L be a frequency square of type F ( n ; λ ) ; that is, an n × n array in which each entry from { 1 , 2 , … , m = n ∕ λ } occurs λ times per row and λ times per column. We show that if m ⩽ 3 , L contains a λ -balanced diagonal, with only one exception up to equivalence when m = 2 . We give partial results for m ⩾ 4 and suggest a generalization of Ryser’s conjecture, that every Latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array. [ABSTRACT FROM AUTHOR]

Abstract: Let n 1 , … , n k ∈ Z + and n 1 + ⋯ + n k = n. The integer partition ( n 1 , … , n k ) of n is said to be realized if there is a latin square of order n with pairwise disjoint subsquares of order n i for each 1 ≤ i ≤ k. In this paper, we construct latin squares realizing partitions of the form ( a s , b t ); that is, partitions with s parts of size a and t parts of size b, where a < b. Heinrich (1982) showed that (1) if s ≥ 3 and t ≥ 3, then there is a latin square realizing ( a s , b t ), (2) ( a s , b ) is realized if and only if ( s − 1 ) a ≥ b, and (3) ( a , b t ) is realized if and only if t ≥ 3. In this paper, we resolve the open cases. We show that ( a 2 , b t ) is realized if and only if t ≥ 3 and ( a s , b 2 ) is realized if and only if a s ≥ b. [ABSTRACT FROM AUTHOR]

We state and prove the first law of cubology of the Rubik's Revenge and provide necessary and sufficient conditions for a randomly assembled Rubik's Revenge to be solvable. [ABSTRACT FROM AUTHOR]

Let T ( n ) denote the maximal number of transversals in an order- n Latin square. Improving on the bounds obtained by McKay et al., Taranenko recently proved that T ( n ) ≤ ( ( 1 + o ( 1 ) ) n e 2 ) n , and conjectured that this bound is tight. We prove via a probabilistic construction that indeed T ( n ) = ( ( 1 + o ( 1 ) ) n e 2 ) n . Until the present paper, no superexponential lower bound for T ( n ) was known. We also give a simpler proof of the upper bound. [ABSTRACT FROM AUTHOR]

We prove that the necessary condition, n ≡ 0 (mod 3), is sufficient for the existence of GDD( n , 2 , 4 ; 3 , 4 ) except possibly for n = 18 . We prove that necessary conditions for the existence of group divisible designs GDD( n , 2 , 4 ; λ 1 , λ 2 ) with equal number of even and odd blocks are sufficient for GDD ( n , 2 , 4 ; 5 n , 7 ( n − 1 ) ) for all n ≥ 2 , GDD ( 7 s , 2 , 4 ; 5 s , 7 s − 1 ) for all s , GDD ( 5 t + 1 , 2 , 4 ; 5 t + 1 , 7 t ) for t ≡ 0(mod 2) and GDD ( 5 t + 1 , 2 , 4 ; 2 ( 5 t + 1 ) , 14 t ) for all t . To complete the existence of such GDDs, one needs to construct two more families: GDD ( 5 t + 1 , 2 , 4 ; 5 t + 1 , 7 t ) for all odd t , and GDD ( 35 s + 21 , 2 , 4 ; 5 s + 3 , 7 s + 4 ) for all positive integers s . [ABSTRACT FROM AUTHOR]

In this paper, some constructions of Yang Hui type magic squares are provided. As their application, it is shown that there exists a Yang Hui type magic square YMS $$(n,2)$$ for all even order $$n\ge 4$$ . Meanwhile, a diagonally ordered irrational YMS $$(n,2)$$ is also mentioned and it is shown that such a YMS $$(n,2)$$ exists for all integers $$n\equiv 2\pmod 4$$ with $$n\ge 6$$ . [ABSTRACT FROM AUTHOR]

A diagonal Latin square is one whose main and back diagonals are both transversals. Difference matrices, quasi-difference matrices and holey quasi-difference matrices are used to construct t pairwise orthogonal diagonal Latin squares of order v ( t PODLS ( v ) ) in this paper. For t = 3 , 4 , 5 , known existence results for t PODLS ( v ) are then updated; for t = 3 , 4, and 5, there remain respectively 1, 11 and 32 unsolved cases, the largest of which are v = 10 , 54 and 98 respectively. [ABSTRACT FROM AUTHOR]

We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota. [ABSTRACT FROM PUBLISHER]

A unified treatment of the 2 × 2 analog of the Freudenthal-Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras. [ABSTRACT FROM AUTHOR]

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0, 1, ⋯, nd} is called a sparse anti-magic square of order n with density d(d

Abstract: In this paper, a strongly symmetric self-orthogonal diagonal Latin square of order with a special property ( SSSODLS( )) is introduced. It is proved that a SSSODLS( ) exists if and only if and . As an application, it is shown that there exists a Yang Hui type magic square YMS if and only if and . [Copyright &y& Elsevier]

Abstract: We show that a pair of orthogonal partial Latin squares of order n can be embedded in a pair of orthogonal Latin squares of order at most and all orders greater than or equal to . This paper provides the first direct polynomial order embedding construction for pairs of orthogonal partial Latin squares. [Copyright &y& Elsevier]

Abstract: In this paper, we show that the minimum number of vertices whose removal disconnects a connected strongly regular graph into non-singleton components equals the size of the neighborhood of an edge for many graphs. These include block graphs of Steiner 2-designs, many Latin square graphs and strongly regular graphs whose intersection parameters are at most a quarter of their valency. [Copyright &y& Elsevier]

We show that two partial latin squares of order mk are simultaneously avoidable if m > 4 and $${k>\frac{m^3(m^2-1)}{2}}$$. If m = 4, we show the same conclusion when k > 56. [ABSTRACT FROM AUTHOR]

Abstract: A spherical latin trade is a partial latin square associated with a face 2-colourable triangulation of the sphere. A latin trade W embeds into an abelian group G if the Cayley table of G contains a partial latin square isotopic to W. A trade W embeds minimally into G if W embeds into G but not into any proper subgroup of G. The canonical group of a spherical latin trade W is a finite abelian group that W embeds into, and possesses a “universal property”: every abelian group that W embeds into minimally is isomorphic to a quotient group of . We study simple graph operations that create larger spherical latin trades from smaller ones. In particular, we analyse the effect of these operations on the canonical groups of the trades. We then apply our results to answer several open problems about canonical and minimal groups recently posed by Cavenagh and Wanless and by Blackburn and McCourt. A family of trades is constructed that exhibits the fastest known asymptotic growth for with respect to the size . It achieves , where . Another family is shown to embed minimally into groups of exponentially increasing order; previously exponential growth had only been observed for canonical groups. Questions about the rank (size of a smallest generating set) of canonical and minimal groups are also resolved. The rank of any minimal group is shown to be at most logarithmic in the size of the trade, resulting in an example where the canonical group grows much faster than any minimal group, with regards to both order and rank. [Copyright &y& Elsevier]

We study the connection between mutually unbiased bases and mutually orthogonal extraordinary supersquares, a wider class of squares which does not contain only the Latin squares. We show that there are four types of complete sets of mutually orthogonal extraordinary supersquares for the dimension d = 8. We introduce the concept of physical striation and show that this is equivalent to the extraordinary supersquare. The general algorithm for obtaining the mutually unbiased bases and the physical striations is constructed and it is shown that the complete set of mutually unbiased physical striations is equivalent to the complete set of mutually orthogonal extraordinary supersquares. We apply the algorithm to two examples: one for two-qubit systems (d = 4) and one for three-qubit systems (d = 8), by using the Type II complete sets of mutually orthogonal extraordinary supersquares of order 8. [Copyright &y& Elsevier]

Abstract: The pandiagonal Latin squares constructed using Hedayat’s method are cyclic. During the last decades several authors have described methods for constructing pandiagonal Latin squares which are semi-cyclic. In this paper we propose a recursive method for constructing non-cyclic pandiagonal Latin squares of any given order , where is a positive composite integer not divisible by 2 or 3. We also investigate the orthogonality properties of the constructed squares and extend our method to construct non-cyclic pandiagonal Sudoku. [Copyright &y& Elsevier]

Abstract: Two Latin squares of order are -orthogonal if their superposition produces exactly distinct ordered pairs. If one of the two squares is the transpose of the other, we say that the square is -self-orthogonal, denoted by . It has been proved by Xu and Chang that the necessary and sufficient condition for the existence of an is and 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., . We also provide an idempotent incomplete self-orthogonal Latin square (ISOLS) of order 26 with a hole of size 8. [Copyright &y& Elsevier]

This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as " our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 핆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 핂 or tensor products 핂1 ⊗ 핂2, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ′, ℍ′, 핆′ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 핂1, 핂2, which in all cases are obtained from ℝ, ℂ, ℂ′, ℍ, ℍ′, 핆, 핆′ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation. [ABSTRACT FROM AUTHOR]

Abstract: Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large. [Copyright &y& Elsevier]

Abstract: Symmetries of a partial Latin square are primarily determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism , the cardinality of the set of partial Latin squares which are invariant under only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties were studied. It is also seen that the cycle structure of determines the possible sizes of the elements of and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square. [Copyright &y& Elsevier]

De La Loubère's method is the most popular rule used to generate magic squares. In this paper we derive a simple formula for the method that is much easier to use and to code than the traditional rule. We also show that the arrays generated from our formula are magic squares, thereby providing a proof of De La Loubère's method. [ABSTRACT FROM AUTHOR]

We use the concept of an availability matrix, introduced in Euler [7], to describe the family of all minimal incomplete 3 x n latin rectangles that are not completable. We also present a complete description of minimal incomplete such latin squares of order 4. [ABSTRACT FROM AUTHOR]

Two orthogonal latin squares of order n have the property that when they are superimposed, each of the n ordered pairs of symbols occurs exactly once. In a series of papers, Colbourn, Zhu, and Zhang completely determine the integers r for which there exist a pair of latin squares of order n having exactly r different ordered pairs between them. Here, the same problem is considered for latin squares of different orders n and m. A nontrivial lower bound on r is obtained, and some embedding-based constructions are shown to realize many values of r. [ABSTRACT FROM AUTHOR]

Abstract: A -way Latin trade of volume is a collection of partial Latin squares , containing exactly the same filled cells, such that, if cell is filled, it contains a different entry in each of the partial Latin squares, and such that row in each of the partial Latin squares contains, set-wise, the same symbols, and column likewise. It is called a -way -homogeneous Latin trade if, in each row and each column, , for , contains exactly elements, and each element appears in exactly times. It is also denoted as a Latin trade, where is the size of the partial Latin squares. We introduce some general constructions for -way -homogeneous Latin trades, and specifically show that, for all , , and , and for all , (except for four specific values), a -way -homogeneous Latin trade of volume exists. We also show that there is no Latin trade and there is no Latin trade. Finally, we present general results on the existence of -way -homogeneous Latin trades for some modulo classes of . [Copyright &y& Elsevier]

Abstract: The sudoku completion problem is a special case of the latin square completion problem and both problems are known to be NP-complete. However, in the case of a rectangular hole pattern–i.e. each column (or row) is either full or empty of symbols–it is known that the latin square completion problem can be solved in polynomial time. Conversely, we prove in this paper that the same rectangular hole pattern still leaves the sudoku completion problem NP-complete. [Copyright &y& Elsevier]

Abstract: A Completely Separating System (CSS) on is a collection of blocks of  such that for any pair of distinct points , there exist blocks such that and . One possible generalization of CSSs are -CSSs. Let be a subset of , the power set of . A point is called -separable if for every -subset there exists a block with and with the property that is disjoint from . If every point is -separable, then is an -CSS (or -CSS). Furthermore, if is a collection of -blocks, then is an -CSS. In this paper we offer some general results, analyze especially the case with the additional condition that , present a construction using Latin squares, and mention some open problems. [Copyright &y& Elsevier]

Abstract: In this paper we provide a construction of normal bimagic squares by means of a magic pair of orthogonal general bimagic squares. It is shown that a normal bimagic square of order exists for all positive integers such that and , and a normal bimagic square of order exists if and only if . [Copyright &y& Elsevier]

The article discusses construction of orthogonal magic sudoku solutions by employing linear magic squares. It mentions that the method employs was based on extension of the method given by the mathematician Simon De la Loubere for constructing magic squares. It states that two sudoku solutions of like order are orthogonal only when each ordered pair of symbols looks exactly same upon superimposition.

Abstract: In this paper, a new concept, -plex orthogonality of Latin squares, is introduced. It generalizes the concept of orthogonality of Latin squares. Some examples of Latin squares with the new orthogonality are given. Bose, Shrikhande, and Parker’s Theorem is generalized to the case of -plex orthogonality for every positive integer while Mann’s Theorem is extended to the case of -plex orthogonality for every positive odd integer . Some other existence or nonexistence theorems are given. We also discuss constructions for -plex orthogonal Latin squares and generalize MacNeish’s Theorem. [Copyright &y& Elsevier]

Abstract: By treating regular (or associative), pandiagonal, and most-perfect (MP) magic squares as matrices, we find a number of interesting properties and relationships. In addition, we introduce a new class of quasi-regular (QR) magic squares which includes regular and MP magic squares. These four classes of magic squares are called “special”. We prove that QR magic squares have signed pairs of eigenvalues just as do regular magic squares according to a well-known theorem of Mattingly. This leads to the fact that odd powers of QR magic squares are magic squares which also can be established directly from the QR condition. Since all pandiagonal magic squares of order 4 are MP, they are QR. Also, we show that all pandiagonal magic squares of order 5 are QR but higher-order ones may or may not be. In addition, we prove that odd powers of MP magic squares are MP. A simple proof is given of the known result that natural (or classic) pandiagonal and regular magic squares of singly-even order do not exist. We consider the reflection of a regular magic square about its horizontal or vertical centerline and prove that signed pairs of eigenvalues of the reflected square differ from those of the original square by the factor i. A similar result is found for MP magic squares and a subclass of QR magic squares. The paper begins with mathematical definitions of the special magic squares. Then, a number of useful matrix transformations between them are presented. Next, following a brief summary of the spectral analysis of matrices, the spectra of these special magic squares are considered and the results mentioned above are established. A few numerical examples are presented to illustrate our results. [Copyright &y& Elsevier]

Abstract: The block graph of a Steiner triple system of order v is a strongly regular graph. For large v, every strongly regular graph with these parameters is the block graph of a Steiner triple system, but exceptions exist for small orders. An explanation for some of the exceptional graphs is here provided via the concept of switching. (Group divisible designs corresponding to) Latin squares are also treated in an analogous way. Many new strongly regular graphs are obtained by switching and by constructing graphs with prescribed automorphisms. In particular, new strongly regular graphs with the following parameters that do not come from Steiner triple systems or Latin squares are found: , , , , , and . [Copyright &y& Elsevier]

Abstract: Using centroskew matrices, we provide a necessary and sufficient condition for a regular magic square to be nonsingular. Using latin squares and circulant matrices we describe a method of construction of nonsingular regular magic squares of order n where n is an odd prime power. [Copyright &y& Elsevier]

The paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we combine the notions of completing and avoiding partial latin squares. Let be a partial latin square of order and let be the set of partial latin squares of order that avoid . We say that is -completable if can be completed to a latin square that avoids . We prove that if has order and contains at most entries, then is -completable for each when . [Copyright &y& Elsevier]

Abstract: An array with entries from is avoidable if there is an Latin square such that no cell in contains a symbol that occurs in the corresponding cell in . We show that the problem of determining whether an array that contains at most two entries per cell is avoidable is -complete, even in the case when the array has entries from only two distinct symbols. Assuming that , this disproves a conjecture by Öhman. Furthermore, we present several new families of avoidable arrays. In particular, every single entry array (arrays where each cell contains at most one symbol) of order with entries from at most distinct symbols and where each symbol occurs in at most cells is avoidable, and every single entry array of order , where each of the symbols occurs in at most cells, is avoidable. Additionally, if , then every single entry array of order at least , where at most rows contain non-empty cells and where each symbol occurs in at most cells, is avoidable. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of two self-orthogonal Latin squares of order (2-). It is found that 2- exist for all except possibly for . 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. [Copyright &y& Elsevier]

Abstract: In this note, a doubly magic rectangle is introduced to construct a doubly pandiagonal magic square. A product construction for doubly magic rectangles is also presented. Infinite classes of doubly pandiagonal magic squares are then obtained, and an answer to problem 22 of [G. Abe, Unsolved problems on magic squares, Discrete Math. 127 (1994) 3] is given. [Copyright &y& Elsevier]

A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies the enumeration of Sudoku grids and an example of the use of this method is given. [ABSTRACT FROM AUTHOR]

A square array is avoidable if for each set of n symbols there is an n × n Latin square on these symbols which differs from the array in every cell. The main result of this paper is that for m ≥ 2 any partial Latin square of order 4 m − 1 is avoidable, thus concluding the proof that any partial Latin square of order at least 4 is avoidable. [ABSTRACT FROM AUTHOR]

Abstract: A Sudoku grid is a 9×9 Latin square further constrained to have nine non-overlapping 3×3 mini-grids each of which contains the values 1–9. In -Quasi-Magic Sudoku a further constraint is imposed such that every row, column and diagonal in each mini-grid sums to an integer in the interval . The problem of proving certain (computationally known) results for concerning mini-grids and bands (rows of mini-grids) was posed at the British Combinatorial Conference in 2007. These proofs are presented and extensions of these provide a full combinatorial enumeration for the total number of completed 2-Quasi-Magic Sudoku grids. It is also shown that there are 40 isomorphism classes of completed 2-Quasi-Magic Sudoku grids. [Copyright &y& Elsevier]

Abstract: A multi-latin square of order and index is an array of multisets, each of cardinality , such that each symbol from a fixed set of size occurs times in each row and times in each column. A multi-latin square of index is also referred to as a -latin square. A -latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in -latin square of order embeds in a -latin square of order , for each , thus generalizing Evans’ Theorem. Exploiting this result, we show that there exist non-separable -latin squares of order for each . We also show that for each , there exists some finite value such that for all , every -latin square of order is separable. We discuss the connection between -latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and -latin trades. We also enumerate and classify -latin squares of small orders. [Copyright &y& Elsevier]

Abstract: Let . We define an equitable Latin rectangle as an matrix on a set of symbols where each symbol appears either or times in each row of the matrix and either or times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of mutually orthogonal equitable Latin rectangles as a . When , or , then we show that the maximum number of for all possible values of . [Copyright &y& Elsevier]

Denote by SFin( v) the set of all integer pairs ( t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure ( t, s). We completely determine the set SFin(2 n) for any integer n ≥ 5. [ABSTRACT FROM AUTHOR]

Abstract: A -plex of a Latin square is a collection of cells representing each row, column, and symbol precisely times. The classic case of is more commonly known as a transversal. We introduce the concept of a -weight, an integral weight function on the cells of a Latin square whose row, column, and symbol sums are all . We then show that several non-existence results about -plexes can been seen as more general facts about -weights and that the weight analogues of several well-known existence conjectures for plexes actually hold for -weights. [Copyright &y& Elsevier]