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Latin Squares

Latin Squares

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Latin squares are combinatorial designs consisting of an n x n array filled with n different symbols, each occurring exactly once in each row and column. They are used in statistical design of experiments, particularly in controlling for two blocking factors.
lightbulbAbout this topic
Latin squares are combinatorial designs consisting of an n x n array filled with n different symbols, each occurring exactly once in each row and column. They are used in statistical design of experiments, particularly in controlling for two blocking factors.

Key research themes

1. How can Mutually Orthogonal Latin Squares (MOLS) be constructed and characterized via algebraic structures such as cellular automata and finite fields?

This theme investigates novel algebraic and combinatorial constructions of MOLS, focusing on linear cellular automata over finite fields and connections to irreducible polynomials. The goal is to characterize when Latin squares generated by algebraic means are orthogonal and to enumerate maximal sets of MOLS for given orders, which is fundamentally important for design theory and related applications in cryptography and coding theory.

Mutually orthogonal latin squares based on cellular automata
by Enrico Formenti

2024, Designs, Codes and Cryptography

Key finding: This paper establishes that Latin squares induced by two Linear Bipermutive Cellular Automata (LBCA) over a finite field F_q are orthogonal if and only if the polynomials associated with their local rules are coprime. It... Read more
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Mutually Orthogonal Latin Squares
by Julian Abel

2025, Chapman and Hall/CRC eBooks

Key finding: The authors prove the existence of four mutually orthogonal Latin squares (MOLS) of order 48, achieving explicit constructive lower bounds for N(v), the maximum number of MOLS of order v, for v < 10,000. This contributes... Read more
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The number of different reduced complete sets of MOLS corresponding to PG $$\varvec{(2,q)}$$ ( 2 , q )
by Jordy Vanpoucke

2023, Journal of Geometry

Key finding: This study calculates the exact number of reduced complete sets of MOLS of order q, where q = p^d is a prime power, corresponding to Desarguesian projective planes PG(2,q). It links the number of such MOLS sets to... Read more
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2. What are the algorithmic and combinatorial methods for generation and random sampling of Latin squares and related designs such as Sudoku?

This theme addresses the development and implementation of efficient algorithms for generating random Latin squares and Sudoku puzzles, focusing on achieving uniform distribution. It involves graphical representations such as incidence cubes, Markov chain Monte Carlo techniques like ±1-moves, and interpretations of Latin squares as maximum cliques in graphs, enabling both enumeration and random sampling. These techniques are essential for applications requiring randomized combinatorial designs, including cryptography, statistical experiments, and recreational mathematics.

Generation of random latin squares step by step and graphically
by Ignacio Gallego Sagastume

2022

Key finding: This work implements the Jacobson and Matthews algorithm for uniformly random generation of Latin squares (specifically order 256), using ±1-moves on incidence cubes — 3D binary arrays encoding symbol placements. The... Read more
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Random Latin squares and Sudoku designs generation
by Roberto Fontana

2021, Electronic Journal of Statistics

Key finding: This paper establishes that Latin squares and Sudoku designs correspond to maximum cliques of suitably constructed graphs and leverages this equivalence to create an algorithm for uniform random sampling of these designs. The... Read more
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Latin squares
by Bhaskar Bagchi

2023, Resonance

Key finding: The authors provide algorithmic procedures to verify if a matrix is a Latin square and enumerative algorithms to generate all Latin squares with first row and column fixed. They analyze the combinatorial explosion in numbers... Read more
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3. How do combinatorial structures such as Latin squares connect to deep algebraic and number-theoretical results, including classical theorems and orbit theory?

This research area explores the fundamental links between Latin squares, number theory, and algebraic structures through combinatorial representation. It includes identifying Latin square identities corresponding to classical sums-of-squares theorems, characterizing prime numbers through orbit structures related to Latin squares, and reflecting on the historical and philosophical significance of these connections. Understanding these links informs both pure mathematics and applied combinatorics, enriching the conceptual framework of Latin squares.

Latin squares and Hurwitz theorem
by Serge Yagunov

2017, Expositiones Mathematicae

Key finding: By establishing a one-to-one correspondence between n-square identities and special types of Latin squares, this paper provides a combinatorial proof of Hurwitz's theorem on sums of squares. This proof uniquely avoids heavy... Read more
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Foundations of Orbit Theory - Definitions and Theorems
by Alan Verdegraal

2022

Key finding: This work formulates Orbit Theory of natural numbers by interpreting numbers via sequences determined by arithmetic progressions modulo n, relating these to Latin square structures with properties such as mirror symmetries... Read more
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Latin Square
by Roger Kirk

2023, Encyclopedia of Social Science Research Methods

Key finding: The entry provides an extensive overview of Latin square properties, equivalence classes, and applications, highlighting their role as multiplication tables of quasigroups and their deep algebraic connections. It elaborates... Read more
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Related Topics

  • Domination in graphs
  • Graph Coloring
  • Combinatorics
  • Pure Mathematics
  • Permutation Arrays
  • Magic Squares
  • Statistics
  • Mathematical Statistics
  • Mathematics
  • Ordinary Differential Equations

    • All papers in Latin Squares

    Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares
    by Shu Lin

    2010

    This paper presents two new large classes of QC-LDPC codes, one binary and one non-binary. Codes in these two classes are constructed by array dispersions of row-distance constrained matrices formed based on additive subgroups of finite... more
    Fig. 4. Error performance of the (4032,3304) QC-LDPC code given in Example 4.
    Fig. 3. Error performances of the (65280,61201) and (65280,32640) QC- LDPC codes given in Example 3.
    Fig. 1. Error performances of the (4032,3708) QC-LDPC code and a (4032,3708) pseudo-random code and a (4032,3708) random MacKay code given in Example 1, where BER and BLER stand for bit and block error rates, respectively
    Fig. 2. Error performances of the algebraic (4032,3276) irregular QC- LDPC code, a (4032,3276) irregular pseudo-random QC-LDPC code and a (4032,3276) random MacKay code given in Example 2.
    Fig. 5. | Error performances of the (4032,3304) QC-LDPC code given in Example 5 decoded using IRB-MLGD, ZF-WBF, JZSC-WBF and SPA algorithms. Hg” add,c,n Satisfies the RC-constraint. For any pair (y,p) of integers with 1 < y,p < q™. let Ham addjen(7, p) be a (7,p) subarray of Hom add,c,n- Hy add,e,n(7; p) is a y(q’” — 1) x p(q’™ — 1) matrix over GF(q™) and it also satisfies the RC-constraint. The null space of Ham add,en(¥, Pp) gives a q™-ary QC-LDPC code Cgm qe over GF(qg™) of length p(q™ — 1) with rate at least (o — y)/p, whose Tanner graph has a girth of at least 6. If Hom addjen(Y, P) does not contain any zero matrix of H 4” add,c,n» it has constant column and row weights, y and p, respectively. Then Cgm gc is a q’-ary (7, p)-regular QC- LDPC code over GF(q). Since Hgm adden (7, 6) satisfies the RC-constraint and each column has y nonzero components (elements in GF(qg™)), a linear sum of any two columns of Hom adden (¥, 6) must result in a column vector with at least 2y — 2 = 2(y — 1) nonzero components (or weight 2y — 2 ), i.e., adding two columns linearly, at most one nonzero component in each column can be canceled. Based on this fact, we readily see that no y or fewer columns of Hy add,cn(¥,5) can be summed to a zero column vector. Consequently, the minimum distance (or minimum weight) of Cqm qc is at least y + 1 (Corollary 3.2.2, [6]). For any given finite field GF(q’”), the above construction gives a family of structurally-compatible q”-ary QC-LDPC codes. Over all the finite fields, the construction gives a large class of q’-ary Sy EL: Pe eee TTne' ~~)... cx: Fe i. ny a Oe, i
    Fig. 6. Block error performances of the 64-ary (1260,630) QC-LDPC code and the (1260,630,631) RS code over GF(2!') given in Example 6.
    COLUMN AND RoW WEIGHT DISTRIBUTIONS OF THE MASKING MATRIX Z(12, 64) IN EXAMPLE 2. 0.2 dB at the BER of 10~®. A random (4032,3708) MacKay code is also constructed for comparison. Its error performance curve with 50 iterations of SPA is on top of that of the pseudo-random (4032,3708) code constructed using the PEG- algorithm as shown in Figure 1. So the algebraic (4032,3708) QC-LDPC code outperforms both random codes in the BER range from 107! to 10~®. a TABLE I
    GENERATORS OF THE CIRCULANTS IN THE REGULAR MASKING MATRIX Z(16, 256) OF EXAMPLE 3
    COLUMN AND RoW WEIGHT DISTRIBUTIONS OF THE IRREGULAR MASKING MATRIX Z(128, 256) IN EXAMPLE 3 10~°, the code performs 0.5 dB from the threshold and 0.55 dB from the Shannon limit. The threshold and the Shannon limit are very close to each other.
    COMPLEXITIES REQUIRED PER ITERATION FOR DECODING AN LDPC CODE WITH VARIOUS ITERATIVE DECODING ALGORITHMS
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    TDMA scheduling design of multihop packet radio networks based on latin squares
    by Victor Li

    1999, IEEE Journal on Selected Areas in Communications

    Absrracr-Many transmission scheduling algorithms have been proposed to maximize the spatial reuse and minimize the Time Division Multiple Access (TDMA) frame length in multihop packet radio networks. Almost all existing algorithms assume... more
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    Constructions of small regular bipartite graphs of girth 6
    by Camino Balbuena

    2011

    In this article, some structures in the projective plane of order q are found which allow us to construct small kregular balanced bipartite graphs of girth 6 for all k ≤ q. When k = q, the order of these q-regular graphs is 2(q 2 −1); and... more
    FIG. 1. Illustration of Subcase t = 1 for g = 4. The deleted points are indicated in white and the deleted incidences by dashed lines.
    FIG. 2. Illustration of Subcase t = 2 for g = 5. The line A, is the curve line, the deleted points are indicated in white, the deleted incidences xjZL,, by dashed lines, and the incidences ajZL, with one black and white point.
    is said to be the position matrix of the family F = {A!, A?, ...,A”}. The following example shows two matri- ces of order 2 x 2 whose elements are subsets of S = {0, a, b} and the position matrix of them. From now on, if there is no confusion, the 1-sets will be indicated without brackets. As symbols, we consider the set [[”]] = {0,1,...,} and denote by (]] = [[7]] \ {0}. Let J, be the identity matrix and denote by FI, the matrix obtained from [, replacing each one with a subset F of (n]]. Let us call array of r symbols and n columns the matrix of order r x n
    Moreover, let O7-t-2, P be the matrix obtained from Og—;-2,4 by changing the entries (Og—;—2,q)uj = u to Zero in every column such that the row j of &/ has q—t symbols different from zero. Thus, for g = 5 and t = 2, we have
    TABLE 1. Order of some small regular graphs of girth 6.
    4. CONCLUSIONS the incidence matrices of k-regular balanced bipartite graphs of girth 6 having 2(qk — 2) vertices where gq is a prime power and k < gq — 1. The following table shows the smallest order n(k, 6) of (k, 6)-graphs achieved until now. In the third row, we cite the reference in which the result has been proved, and by (*), we denote a result in this article. First, for all integers g such that there exists a projec- tive plane of order g, we find k-regular balanced bipartite graphs of girth 6 with 2(qk — 2) vertices fork < q— 1 by using geometrical methods. Taking one step further, we have presented a method based on Latin squares providing
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    Research problems from the BCC22
    by Peter Cameron

    2011, Discrete Mathematics

    The Research Problems section presents unsolved problems in discrete mathematics. The problems here were posed at the problem session of the 22nd British Combinatorial Conference. Problems subsequently solved have been removed. The... more
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    Generalized Hadamard matrices and colourable designs in the construction of regular GDDs with two and three association classes
    by Jennifer Seberry

    1986, Journal of Statistical Planning and Inference

    Generalized Hadamard matrices and colourable designs are used to construct many new group divisible designs. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J, Generalized Hadamard matrices and colourable... more
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    Group divisible designs with three groups and block size four
    by dinesh sarvate

    2007, Discrete Mathematics

    We present new constructions and results on GDDs with three groups and block size four and also obtain new GDDs with two groups of size nine. We say a GDD with three groups is even, odd, or mixed if the sizes of the non-empty... more
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    Fractional latin squares, simplex algebras, and generalized quotients
    by Jerzy Wojciechowski

    2000, Journal of Statistical Planning and Inference

    Recently the authors deÿned the concept of a weighted quasigroup, and showed that each weighted quasigroup is the amalgamation of a quasigroup. Similar results were obtained for symmetric and other types of quasigroups.
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    Orthogonally blocked mixture component–amount designs via projections of F-squares
    by Poonam Singh

    2012, Journal of the Korean Statistical Society

    Orthogonal block designs for Scheffé's quadratic model have been considered previously by Draper et al. (1993), John (1984), Lewis et al. (1994) and Prescott, Draper, Dean, and Lewis (1993). Prescott and Draper (2004) obtained mixture... more
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    Some new constructions of strength 3 mixed orthogonal arrays
    by Man Nguyen

    2008, Journal of Statistical Planning and Inference

    Orthogonal arrays of strength 3 permit estimation of all the main effects of the experimental factors free from confounding or contamination with 2-factor interactions. We introduce methods of using arithmetic formulations and Latin... more
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    Completing Latin squares: Critical sets II
    by Italo Dejter

    2007, Journal of Combinatorial Designs

    It is shown that each critical set in a Latin square of order n > 6 has to have at least 7n− √ n−20 2 empty cells.
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    Design of Strong Cryptographic Schemes based on Latin Squares
    by Reshu Chaudhary
    A Latin Square (LS) of order n is an arrangement of n symbols in an nxn matrix form so that each symbol occurs in each row and each column exactly once. The total number of Latin Squares LS(n) of order n increases rapidly with n. This... more
    2.1 Definition : A Latin square [5, 6] of order n is an n x n square matrix whose entrie: consist of n symbols such that each symbol appears exactly once in each row and eact column. Examples of a 4 x 4 and 5 x 5 Latin square are given below 2.2 Reduced/Standard Latin Squares: Given a Latin square we can permute the columns so that the first row consists of 1, 2..., n in the natural order. After doing this we could permute the rows so that the first column is also 1, 2..., n in the natural order. The resulting square is called Reduced / Standard Latin square.
    We first report observations for the original QG encryption algorithm with inputs as (a) a sequence ‘AA ... A’ and (b) a normal English text sequence taken from the daily news.
    RESULT OF ORIGINAL ALGORITHM WITH ENGLISH TEXT SEQUENCE ies
    ESULT OF IMPROV] ED ALGORITHM WITH | ENGLISH TEXT SI EQUENC] Uy
    RESULT OF ORIGINAL ALGORITHM WITH | ENGLISH TI EXT SI QUI ENC] Uy
    ESULT OF IMPROV] ED ALGORITHM WITH | ENGLISH T] EXT SI QUI ENCE
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    Random Latin squares and Sudoku designs generation
    by Roberto Fontana

    2014, Electronic Journal of Statistics

    Uniform random generation of Latin squares is a classical problem. In this paper we prove that both Latin squares and Sudoku designs are maximum cliques of properly defined graphs. We have developed a simple algorithm for uniform random... more
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    The minimum size of critical sets in latin squares
    by Hung-lin Fu

    1997, Journal of statistical planning and inference

    A critical set C of order n is a partial latin square of order n which is uniquely completable to a latin square, and omitting any entry of the partial latin square destroys this property. The size s(C) of a critical set C is the number... more
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    Adaptable Probabilistic Transmission Framework for Wireless Sensor Networks
    by Vladimir Zadorozhny

    2009, 2009 Third International Conference on Sensor Technologies and Applications

    We propose a novel framework that combines probabilistic transmission with Latin Squares characteristics to tune channel access, meeting various demands in network performance (Energy vs. Delay). The proposed technique is decentralized,... more
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    On quasigroups with few associative triples
    by Otokar Grosek

    2012, Designs, Codes and Cryptography

    There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower... more
    2 Values of a(n) for small n For 1 <n < 6, the values of a(n) have been determined in Jezek-Kepka [3]. Using a1 >xhaustive computer search we found the value of a(7). In addition, we have obtained a1 upper bound on a(8). With respect to Theorem 1.1, we looked for an idempotent quasi croup with the minimum number of associative triples. Our computational experiments d not provide any evidence that the extremal quasigroups should be idempotent. Therefore 11 the future we plan to seek a lower bound on a(n) that will not include the invariant i(Q) The left column of Table | contains the values of a(n) while the right column contains thi minimum number of associative triples over all idempotent quasigroups. Tn [4] the anthoare nrecent an evampnle afidemnnatent qnacoraimn OO af arder 7 with v() —- Table 1 Extremal values of a(Q) for general and idempotent quasigroups
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    A new class of facets for the Latin square polytope
    by Ioannis Mourtos

    2006, Discrete Applied Mathematics

    Latin squares of order n have a 1-1 correspondence with the feasible solutions of the 3-index planar assignment problem (3PAP n). In this paper, we present a new class of facets for the associated polytope, induced by odd-hole inequalities.
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    On nonsingular regular magic squares of odd order
    by Michael Z. Lee

    2012, Linear Algebra and its …

    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... more
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    On Non-Polynomial Latin Squares
    by Otokar Grosek

    2000, Designs, Codes and Cryptography

    It turns out that Latin squares which are hard to approximate by a polynomial are suitable to be used as a part of block cipher algorithms (BCA). In this paper we state basic properties of those Latin squares and provide their construction.
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    Tight bounds for break minimization in tournament scheduling
    by Gerhard Post

    2008, Journal of Combinatorial Theory, Series A

    We consider round-robin sports tournaments with n teams and n − 1 rounds. We construct an infinite family of opponent schedules for which every home-away assignment induces at least 1 4 n(n − 2) breaks. This construction establishes a... more
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    A Quantitative Approach to Perfect One-Factorizations of Complete Bipartite Graphs
    by Natacha Astromujoff

    The Electronic Journal of Combinatorics

    Given a one-factorization $\mathcal{F}$ of the complete bipartite graph $K_{n,n}$, let ${\sf pf}(\mathcal{F})$ denote the number of Hamiltonian cycles obtained by taking pairwise unions of perfect matchings in $\mathcal{F}$. Let ${\sf... more
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    Latin squares and Hurwitz theorem
    by Serge Yagunov

    2015, Expositiones Mathematicae

    We give a combinatorial proof of Hurwitz theorem on the sums of squares for polynomials with integer coefficients. It is shown that n-square identities are in one-to-one correspondence with special types of Latin squares. This enables us... more
    Lemma 4. Every normalized well-painted square splits in 2 x 2 blocks. Each block has either the form
    In the table below one can see an example of a polarized square of order eight containin; polarized subsquares of all possible smaller orders. are labeled with an odd number of + (and —) labels. Firstly, one has:
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    Generation of Random Latin Squares Step By Step and Graphically
    by Ignacio Gallego Sagastume

    2014, Generation of Random Latin Squares Step By Step and Graphically

    In order to generate random Latin squares of order 256 [13], the Jacobson and Matthews' algorithm [1] has been implemented in Java. Clear and efficient data structures (for squares and incidence cubes) have been modeled and the ±1-moves... more
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    EXPERIMENTAL INVESTIGATION ON CIRCULAR HOLLOW STEEL COLUMNS IN-FILLED WITH LIGHT WEIGHT CONCRETE (WITH & WITHOUT GFRP) UNDER CYCLIC LOADING
    by Editor IJRET
    Composite Circular hollow Steel tubes with and without GFRP infill for three different grades of Light weight concrete are tested for ultimate load capacity and axial shortening , under Cyclic loading. Steel tubes are compared for... more
    Recently, composite columns are finding a lot of usage for seismic resistance. In order to prevent shear failure of RC column resulting in storey collapse of building, it is necessary to make ductility of columns larger. Recently, most of building utilizes this Concrete Filled tubes (CFT) concept as primary for lateral load resisting frames. The concrete used for encasing the structural steel section not only enhances its strength & stiffness, but also protects it from fire damages Columns occupy a vital place in the structural system. Weakness or failure of a column destabilizes the entire structure. Strength & ductility of steel columns need to be ensured through adequate strengthening, repair & rehabilitation techniques to maintain adequate structural performance.
    Where the gj's are properly chosen continuous functions of one variable, and the ij's are continuous monotonically increasing functions independent of f. Any continuous real-valued functions f (x1, x2, ..., xn) defined on[0, 1Jn, # =2, canbe represented in the form
    Fig2 (a): Block diagram of feed forward network
    An elementary neuron with R inputs is shown below. Each input is weighted with an appropriate w. The sum of the weighted inputs and the bias forms the input to the transfer
    2.4 Train the Network
    Fig4. 200T Cyclic Testing Machine 6. EQUIPMENTS & EXPERIMENTAL PROGRAM
    Figo. Specimens failure after testing Load vs Length & Stress vs Strain
    IC-RICE Conference Issue | Nov-2013, Available @ http://www. ijret.org
    Table2. Comparison of Pin, Purxp & Pann Figure6. Comparison of Pin, Purxp & Pann
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    Generación de Cuadrados Latinos de Orden 256 Utilizando un Grafo de Reemplazos
    by Ignacio Gallego Sagastume

    2015, Generación de Cuadrados Latinos de Orden 256 Utilizando un Grafo de Reemplazos

    Los cuadrados Latinos (LSs) son estructuras algebraicas con aplicaciones en criptografía. Si los LSs son aleatorios y uniformemente distribuidos, pueden ser usados como claves para algoritmos de encriptación simétricos. En el contexto de... more
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    Covering radius for sets of permutations
    by Peter Cameron

    2005, Discrete Mathematics

    We study the covering radius of sets of permutations with respect to the Hamming distance. Let f (n, s) be the smallest number m for which there is a set of m permutations in S n with covering radius r ≤ n − s. We study f (n, s) in the... more
    The last inequality relies on the assumption that n > 2s + 3. Choose a fixed permutation h € S,,. If x is a random permutation, the proba- bility that h is not within distance n—s—1 of x is thus q. Hence, if 71,...,%m are independent random permutations, the probability that h is not within distance n — s — 1 of any of them is q™. So the expected number of permuta- tions uncovered by m balls of radius n — s — 1 with random centres is n! q". If this is less than 1, then there is a set of points of cardinality m with covering radius at least n — s, and so f(n,s) <m. Taking m = e(s+1)!nlogn we find from (1) that mlogqg < —nlogn and hence n! q™ < 1 as required.
    The values for s = 1 and s > n — 1 follow from our earlier remarks, while f (4,2) = 4 comes from Proposition 4 and Corollary 6. The entry 12..20 means that the value of f(5, 3) is in the range 12 to 20. The lower bound was proved above, and the upper bound follows from the fact that the group AGL(1, 5) of order 20 has covering radius 2 (shown in the next section), or from the bound of Proposition 11. A computation using GAP [11] shows that m(5,1) = 7, giving f(5,2) < 7. (By contrast, the largest (<1)-intersecting subset of S; has size 20. We use the GAP share package GRAPE [16] to find all cliques in the graph G52, up to automorphisms of the graph.) The correct value of f(5,2) is 6. The lower bound is found by brute force, and the upper bound comes from the example in the preceding subsection.
    Fig. 1. Counter-example to Derienko’s method of B. After that we make the equivalent steps among the alphabetic symbols. After 12 more steps we are back in the bottom right, undoing the steps we first made there. Finally, we go back to the top left and undo the steps we made there. So, after four lots of 12 perturbations we are back to where we started.
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    Estructuras algebraicas aplicables en criptografía
    by Ignacio Gallego Sagastume

    2016, Estructuras algebraicas aplicables en criptografía

    Existen diferentes estructuras algebraicas que son muy útiles en el ámbito de la seguridad informática y en particular en aplicaciones criptográficas. Algunas de estas estructuras son los cuadrados Latinos (CLs o LSs) y los quasi-grupos... more
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    Generation of Random Latin Squares Using a Random-Swapping Technique
    by Ignacio Gallego Sagastume

    Generation of Random Latin Squares Using a Random-Swapping Technique

    Random Latin squares are useful in cryptography, in order to construct ciphers for cryptographic protocols ([12,3]), or dierent cryptographic applications. Those of order 256 are of special interest, because the entire ASCII table can be... more
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