We give a new construction, for all primep⩾11, of power sets of latin squares not based on group tables. We discuss some conjectures about latin power sets.

Foreword (P. Erdos). Introduction (J. Denes, A.D. Keedwell). Transversals and Complete Mappings (J. Denes, A.D. Keedwell). Sequenceable and R-Sequenceable Groups: Row Complete Latin Squares (J. Denes, A.D. Keedwell). Latin Squares With and Without Subsquares of Prescribed Type (K. Heinrich). Recursive Constructions of Mutually Orthogonal Latin Squares (A.E. Brouwer). r-Orthogonal Latin Squares (G.B. Belyavskaya). Latin Squares and Universal Algebra (T. Evans). Embedding Theorems for Partial Latin Squares (C.C. Lindner). Latin Squares and Codes (J. Denes, A.D. Keedwell). Latin Squares as Experimental Designs (D.A. Preece). Latin Squares and Geometry (J. Denes, A.D. Keedwell). Frequency Squares (J. Denes, A.D. Keedwell). Bibliography. Subject Index.

A branch and bound algorithm for the planar three-index assignment problem is presented. The branching operation is performed on sets of variables belonging to a common constraint rather than on a single variable. The algorithm comprises an upper and lower bound procedure. The upper bound procedure consists of two parts: one for finding an initial solution and one for improving it. The local improvement rule takes advantage of the Latin square structure of the solution. The lower bound procedure combines a new dual heuristic and a Lagrangean relaxation scheme solved by subgradient optimization. The use of reduced costs in the evaluation of bounds is another feature of the algorithm. Computational experience both on the exact algorithm and on the primal heuristics as standalone procedures is reported.

Food systems need to become more sustainable. There is a need to investigate the agricultural management components that address the sustainability better. Long crop rotations are suggested to be environmentally friendly, yet, little is known how soil microbial communities may be affected by long-term rotation under organic cropping with cover crops and manure and conventional cropping with different nitrogen rates. We examined the composition and diversity of soil bacterial and fungal communities in a five-field crop rotation at the beginning and end, respectively in 2013 and 2018. Our analysis revealed that bacterial and to a lesser extent fungal diversity increased by the end of the rotation in all organic treatments and in conventional treatments with low to medium nitrogen rate (20‐100 kg of nitrogen per hectare). Conventional treatment with no added nitrogen decreased bacterial and fungal diversity. Nitrogen rate of 150 kg/ha decreased only bacterial diversity, while the impact on fungal diversity was neutral. Crop rotation significantly increased the relative abundance of bacterial taxa involved in nitrification and denitrification. Of fungal functional groups, the relative abundance of pathogenic functional groups decreased and mycorrhizal groups increased during crop rotation and especially with added cover crops. Our results suggest that crop rotation may outperform cropping systems in structuring soil microbial communities.

In a recent book (Elements of Digital Satellite Communication, vol.II, 1985), W.W. Wu has shown that all the secret-sharing systems known at the time his book was written can be described in terms of Reed-Solomon codes. He has also remarked that all are connected with latin squares and has illustrated this remark by means of examples. However, his examples construct secret-sharing systems in which the secret can be unlocked with only two keys. It is shown here that this is necessarily the case when latin squares are used. For this purpose. Wu's observations are replaced by a theorem, one which also shows a connection with Golomb-Posner codes.< >

A finite group is said to be admissible if it has a permutation mapping of the form g → θ( g ) such that g → g.θ( g ) is also a permutation. Two different necessary conditions for a group to be admissible are known. For abelian groups both are sufficient and for soluble groups at least one. Here, we show that both conditions are satisfied by all finite non-soluble groups and so we conjecture that all such groups may be admissible.

In this column Periodica Mathematica Hungarica publishes current research problems whose proposers believe them to be within the reach of existing methods. Manuscripts should preferably contain the background of the problem and all references known to the author. The length of the manuscripts should not exceed two double-space type-written pages.

(1988). A Conjecture Concerning Magic Squares. The American Mathematical Monthly: Vol. 95, No. 9, pp. 845-849.

Several ways of allocating frequencies efficiently are suggested that are based on the use of Latin squares. A Latin square of order n is a square array of n rows and n columns, and involving n symbols each of which occurs n times within the square in such a way that all the different symbols occur once in each row and once in each column. This approach provides two alternative ways of arranging the transmitters for a mobile radio telephone system, one rectangular and the other close-packed hexagonal. In both cases optimal patterns for frequency allocation result. A solution using the least-possible number of frequencies for each case is given, as well as one that is ideal in the sense that all the transmitters that surround any particular one are provided with distinct frequencies. >

Several ways of allocating frequencies efficiently are suggested that are based on the use of Latin squares. A Latin square of order n is a square array of n rows and n columns, and involving n symbols each of which occurs n times within the square in such a way that all the different symbols occur once in each row and once in each column. This approach provides two alternative ways of arranging the transmitters for a mobile radio telephone system, one rectangular and the other close-packed hexagonal. In both cases optimal patterns for frequency allocation result. A solution using the least-possible number of frequencies for each case is given, as well as one that is ideal in the sense that all the transmitters that surround any particular one are provided with distinct frequencies. >