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On the Existence of De Bruijn Tori with Two by Two Windows On the Existence of de Bruijn Tori with Two by Two Windows Glenn Hurlb ert Department of Mathematics Arizona State University Tempe Arizona USA Chris J Mitchell Department of Computer Science Royal Holloway University of London Egham Surrey TW EX UK and Kenneth G Paterson Department of Mathematics Royal Holloway University of London Egham Surrey TW EX UK May This authors research supp orted by a Lloyds of London Tercentenary Foundation Research Fellowship Abstract Necessary and sucient conditions for the existence of de Bruijn Tori or Perfect Maps with two by two windows over any alphab et are given This is the rst twodimensional window size for which the existence question has b een completely answered for every alpha b et The techniques used to construct these arrays utilise existing results on Perfect Factors and Perfect MultiFactors in one and two dimensions and involve new results on Perfect Factors with punctur ing capabilities Finally the existence question for twodimensional Perfect Factors is considered and is settled for two by two windows and alphab ets of primep ower size Introduction A dimensional C ary de Bruijn Torus or Perfect Map is a dimensional p erio dic array with symbols drawn from an alphab et of size C having the prop erty that every C ary array of some xed size o ccurs exactly once as a subarray of the array More precisely in the notation of an R S u v C dBT is an R S p erio dic array with symbols drawn from a set of size C having the prop erty that every p ossible u v array o ccurs exactly once as a p erio dic subarray of the array The pair u v is often called the window of the torus A variety of notations have b een introduced to describ e these arrays we use that of Perfect Factors are related ob jects which have proved useful in construc u C tions for de Bruijn Tori an R u T PF is a set of T C ary p erio d R C R sequences in which every C ary utuple o ccurs exactly once as a subsequence The parameter u is often called the span of the sequences Perfect Factors have b een extensively studied in Both de Bruijn Tori and Perfect Factors are generalisations of the classical de Bruijn sequences a de Bruijn sequence is a Perfect Factor with T ie u a C u PF while de Bruijn Tori are twodimensional analogues of de C u Bruijn sequences We denote a C ary span u de Bruijn sequence by C u C u dBS It was proved in that there exists a C u dBS for all integers C C and u As noted in there are many applications for such ob jects In two dimensions one can dene an R S u v T PF as a set of T C R S p erio dic arrays with symbols drawn from a set of size C having the prop erty that every p ossible u v array o ccurs exactly once as a p erio dic subarray in precisely one of the arrays Of course an R S u v PF is C simply an R S u v dBT C The following necessary conditions on the parameters of an R S u v C dBT are easily established Lemma Suppose there exists an R S u v dBT Then C uv i RS C ii R u or R u iii S v or S v It has b een conjectured that these conditions are in fact also sucient for the existence of de Bruijn Tori This has b een proved in a wide variety of cases and we outline the relevant results next In contrast very little is known ab out the existence of twodimensional Perfect Factors Combining ideas from and the conditions of Lemma were shown to b e sucient for the existence of an R S u v dBT in This work 2 was extended to alphab ets of primep ower size in Square de Bruijn Tori ie R R u u dBT were considered in while some families of arrays C with u v were constructed in The existence question for de Bruijn Tori over general alphab ets was studied in and higher dimensional versions were considered in The main result of is as follows Supp ose C has prime factorisation k Y c i C p i i=1 uv By Lemma RS C and so any prime dividing R or S also divides C We can therefore write k k Y Y r s i i R p S p i i i=1 i=1 for some r c uv where s c uv r R u and S v i i i i i Result Theorem Suppose C R and S have prime factorisations as above and that for some i we have r s i i p u and p v i i Then there exists an R S u v dBT C The parameter sets satisfying Lemma for which existence of de Bruijn r s i i u or p Tori remains unsettled after Result are those where either p v i i r s i i for each i The cases where either p u for every i or p v for every i i i could b e resolved if the existence question for onedimensional Perfect Factors were p ositively settled by using the generalisation of Etzions construction given in Recent progress on this problem can b e found in r i However there would still remain the mixed parameter sets where p u i s i for some indices and p v for other indices Our purp ose in this pap er i is to develop new construction metho ds for these mixed parameters in the simplest nontrivial case where u v We are then able to settle the existence question for de Bruijn Tori with u v In a mixed parameter set with u v one of the following two cases must o ccur for each i p and r c or c i i i i p and r or c i i i The case where every r is either or c is covered by the following result i i which we prove in Section partially answering a question of 4 4 Theorem Suppose m n Then there exists an m n dBT mn We consider another sub case in Section where p r and r c 1 1 i i for i Our main result in that section is Theorem Suppose n is odd Then for every c there exists a 4 4c1 c n dBT 2 n The pro ofs of Theorems and rest on the construction of some sp ecial classes of onedimensional Perfect Factors in Section The sequences of these Factors have what we call puncturing capabilities a generalisation of the notion of the puncturing of de Bruijn sequences In Section we combine Theorems and with some results from to obtain Theorem The necessary conditions of Lemma are sucient for the ex istence of an R S dBT C Theorem provides the rst instance where necessary and sucient con ditions are known for the existence of de Bruijn Tori with a xed two dimensional window size over all alphab ets Having solved the existence problem for de Bruijn Tori with windows we go on to investigate the analogous problem for R S T PFs in C Section We obtain a complete answer in the case where C is a prime p ower Theorem Let p be a prime and c r s and t be integers The conditions r s that p p and r s t c are necessary and sucient for the existence r s t c of a p p p PF p We close with some op en questions Some Classes of Perfect Factors Perfect MultiFactors We introduce a class of combinatorial ob jects which have proved useful in the construction of Perfect Factors In order to unify the presentation we use notation dierent from that found in Additionally we say that a v tuple appears in a sequence at position 0 1 v 1 0 1 z if z z +1 z +v 1 0 1 v 1 Denition Suppose m n C and u are positive integers which sat u isfy mjC and C An R u T n Perfect MultiFactor or simply a C u R u T nPMF is a set of T C m C ary sequences of period R mn C with the property that for every C ary utuple and integer j with j n occurs at a position z with z j mo d n in one of the sequences Result Theorem Suppose n C u are positive integers C u and n u Then there exists an n u C nPMF containing the al lzero C sequence Lemma Suppose n C l are positive integers with n C Let z 0 z z z n be a sequence of integers with z z for j 1 l1 l j +1 j 2 l Then there exists an n C nPMF with sequences C i j i j C having the property that sequence i j of the PMF satises i j i j i j i z z z 0 1 l1 we say that i j is constant in positions z z z Moreover when 0 1 l1 l C the PMF can be constructed so that sequence i j also satises i j i j i z z +1 j j we say that i j is constant in positions z and z j j Proof Since R z z for j l there exists for each j j +1 j 2 j j an R C R PMF A as in Result Our aim is to concatenate j C j j sequences from each A to form sequences of p erio d n which comprise the required PMF j j Let a b denote the unique sequence in A which b egins with the j tuple a b and let L a b denote the last symbol of that sequence Let j j j X f i b b C g Because of the dening prop erty of A the list i j j j L i L i L i c j consisting of the nal symbols of the sequences in X is a p ermutation i 0 of f C g Thus concatenating each sequence in X with each i 1 2 sequence of X in any pairing for each i pro duces a set of C sequences i 2 that constitute a z C z PMF 2 C 2 More generally if we denote the concatenation of a list of l sequences i l by l and if is a p ermutation of j f C g for each i C and each j l we have 2 that the C sequences l1 i 1 i 0 i j i j C j i j i i j i l1 1 0 2 constitute an n C nPMF It is clear that the sequences i j satisfy C i j i j i j i so that the rst condition in the z z z 0 1 l1 statement of the lemma holds i for i Now supp ose that l C In dening the p ermutations j C and j l we additionally sp ecify that i j i i j C j Then tuple i i app ears in sequence i j at p osition z and the second j condition in the statement of the lemma is satised 2 Puncturing and Joining Sequences Let K denote the complete directed graph with lo ops on m vertices An m mary span de Bruijn
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