If $D$ is a $(4u^2,2u^2-u,u^2-u)$ Hadamard difference set (HDS) in $G$, then
$\{G,G\setminus D\}$ is clearly a $(4u^2,[2u^2-u,2u^2+u],2u^2)$ partitioned
difference family (PDF). Any $(v,K,\lambda)$-PDF will be said of Hadamard-type
if $v=2\lambda$ as the one above. We present a doubling construction which,
starting from any such PDF, leads to an infinite class of PDFs. As a special
consequence, we get a PDF in a group of order $4u^2(2n+1)$ and three
block-sizes $4u^2-2u$, $4u^2$ and $4u^2+2u$, whenever we have a
$(4u^2,2u^2-u,u^2-u)$-HDS and the maximal prime power divisors of $2n+1$ are
all greater than $4u^2+2u$

Buratti, Marco

English

arXiv

If D is a (4u2, 2u2 − u, u2 − u) Hadamard difference set (HDS) in G, then {G,G  D} is clearly a (4u2, [2u2 − u, 2u2 + u], 2u2) partitioned difference family (PDF).
Any (v,K, λ)-PDF will be said a Hadamard PDF if v = 2λ as the one above. We present a doubling construction which, starting from any Hadamard PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order 4u2(2n + 1) and three block-sizes 4u2 − 2u, 4u2 and 4u2 + 2u, whenever we have a (4u2, 2u2 − u, u2 − u)-HDS and the maximal prime power divisors of 2n + 1 are all greater than 4u2 + 2u

Marco Buratti

Archivio della ricerca- Università di Roma La Sapienza

https://doi.org/10.1007/s12095-018-0308-3Hadamard partitioned difference families and theirdescendantsMarco Buratti1Received: 17 May 2017 / Accepted: 19 April 2018© Springer Science+Business Media, LLC, part of Springer Nature 2018Abstract If D is a (4u2, 2u2 − u, u2 − u) Hadamard difference set (HDS) in G, then{G, G \ D} is clearly a (4u2, [2u2 − u, 2u2 + u], 2u2) partitioned difference family (PDF).Any (v,K, λ)-PDF will be said a Hadamard PDF if v = 2λ as the one above. We present adoubling construction which, starting from any Hadamard PDF, leads to an infinite class ofPDFs. As a special consequence, we get a PDF in a group of order 4u2(2n + 1) and threeblock-sizes 4u2 − 2u, 4u2 and 4u2 + 2u, whenever we have a (4u2, 2u2 − u, u2 − u)-HDSand the maximal prime power divisors of 2n + 1 are all greater than 4u2 + 2u.Keywords Partitioned difference family · Hadamard difference set ·Strong difference familyMathematics Subject Classification (2010) 05B10 · 94B251 IntroductionThroughout this note the multiset sum of two multisets X and Y on a set S is the multisetX  Y where the multiplicity of every element of S is the sum of its multiplicities in Xand Y . The multiset sum of μ copies of a multiset X will be denoted by μX. The multisetμ1{x1}  . . .  μt {xt } will be denoted by [μ1x1, . . . , μt xt ].Given an additive group G and a multiset X = {x1, . . . , xk} on G, the list of differencesof X is the multiset X of all possible differences xi − xj with (i, j) an ordered pair ofdistinct elements in {1, . . . , k}. More generally, the list of differences of a collection X of Marco Burattiburatti@dmi.unipg.it1 Dipartimento di Matematica e Informatica, Università di Perugia,via Vanvitelli 1, Perugia, 06123, Italy/ Published online:  2018Cryptogr. Commun. (2019) 11: –5 5April25 57 62multisubsets of G is the multiset sum X :=⊎X∈XX. Assume that K is the multiset ofsizes of the members of X . The following cases are important:• X = λ(G \ H) for a suitable λ and a suitable subgroup H of G. Here one says thatX is a (G,H, K, λ) difference family (DF).• X = μG for a suitable μ. Here one says that X is a (G,K,μ) strong differencefamily (SDF).Very often one refers to a (G,H, K, λ)-DF as a (v, h,K, λ) difference family in G rel-ative to H where v and h denote the orders of G and H , respectively. One speaks of anordinary difference family when H = {0}. In this case one simply writes (G,K, λ)-DF or(v, K, λ)-DF in G.The members of a DF or SDF are called blocks. It is clear that every block of a DF is a setwhile a SDF must have at least one block with repeated elements. A DF or SDF with onlyone block is said to be a difference set (DS) or a difference multiset (also called a differencecover in [1, 14]), respectively. If one writes k instead of K , it means that all blocks havesize k.Partitioned difference families (PDFs), also known as zero difference balanced functions(see, e.g., [17, 19]), are (G,K, λ)-DFs whose blocks partitionG. They have been introducedby Ding and Yin [10] for the construction of optimal constant composition codes. They arealso important from the design theory perspective; for instance, it is shown in [6] that a PDFwith K = [1(k − 1), rk] gives rise to a resolvable 2-design with block size k.In the class of PDFs we have, in particular, the (G, [ 11, nk], k − 1)-PDFs giving riseto sharply point transitive near resolvable 2-(kn + 1, k, k − 1) designs. There are manyconstructions of PDFs of this kind in the literature. Anyway, as shown in [7], most of themare encompassed in a unique simple algebraic construction.In this paper we are interested in (v,K, λ)-PDFs having v = 2λ. The motivation willbe given by our main result: each PDF with this property leads to an infinite class ofPDFs obtainable by means of a new doubling construction. We point out that, so far, thecomposition constructions for PDFs essentially made use of difference matrices [6, 13].Definition 1 A Hadamard PDF is a (G,K, λ)-PDF with |G| = 2λ.We choose the name “Hadamard” since every Hadamard difference set (HDS) immedi-ately gives a PDF with the required property. For convenience of the reader we recall that aHDS is a difference set of parameters (4u2, 2u2 − u, u2 − u) for some u (see, e.g., [12]).Proposition 1 If D is a HDS in G, then {D, G \ D} is a Hadamard PDF.Proof It is known and trivial that if D is a (v, k, λ)-DS in G, then {D, G \ D}is a(v, [k, v−k], v−2k+2λ)-PDF in G. So, in particular, if D is a(4u2, 2u2−u, u2−u)-HDS,then {D,G \ D} is a (4u2, [2u2 − u, 2u2 + u], 2u2)-PDF in G. The assertion follows.At the moment to find another class of Hadamard PDFs seems to be quite hard to thisauthor. Anyway the set of Hadamard PDFs not coming from the above proposition is notempty. Indeed we have found a (32, [22, 6, 22], 16)-PDF in the non-abelian group G whoseelements are all pairs of the Cartesian product Z4 × Z8 and whose operation law is(x1, y1) + (x2, y2) = (x1 + x2, 5x2y1 + y2).5 85 Cryptogr. Commun. (2019) 11: –5 557 62One can check that the four blocks of the mentioned PDF are the following:X1 = {(0, 0), (2, 0)}; X2 = {(1, 0), (3, 4)}X3 = {(0, 1), (0, 3), (1, 2), (1, 5), (1, 6), (3, 3)}; X4 = G \ (X1 ∪ X2 ∪ X3).2 Hadamard strong difference familiesStrong difference families have been formally introduced in [3] but they have been implic-itly used in many earlier papers. Note that they have no known relation to the “strongexternal difference families” very recently introduced in [15], in spite of the misfortune ofinadvertently similar terminology.Strong difference families might be very useful to construct relative difference families[4, 5, 8, 9, 14]; in a certain sense they are the “skeleton” of the resulting difference family asshown in the following construction which is a little bit more general than the “fundamentalconstruction” in [3].Theorem 1 Let X = {X1, . . . , Xt } be a (G,K,μ)-SDF. Take a group H and a pair ofpositive integers (n, λ) with μn = λ(|H | − 1). Then take a collection B = {B1, . . . , Bt } ofsubsets of G × H such that the projection of Bi on G coincides with Xi for i = 1, . . . , t .We haveB =⊎g∈G{g} × Lgfor suitable lists Lg of elements of H . Here, by definition of a SDF, every Lg has con-stant size μ. This is important because it allows for the possible existence of an n-set E ofendomorphisms of H for which we have⊎ε∈Eε(Lg) = λ(H \ {0}) ∀g ∈ G.If such a set E exists, extend each ε ∈ E to the endomorphism ε̂ of G × H defined byε̂(g, h) = (g, ε(h)) for each (g, h) ∈ G × H . Then {̂ε(Bi) | 1 ≤ i ≤ t; ε ∈ E} is a(G × H, G × {0}, nK, λ)-DF.The “fundamental construction” in [3] corresponds to the case that H is the additivegroup of a finite field Fq and each ε ∈ E is the multiplication by a suitable non-zero elementof the field. Thus, it is in some way related with the “factorization of a group” [16] and,more generally, with the “multifold factorization of a group” [2, 18]. Indeed the constructionsucceeds if there exists E ⊂ F∗q such that E · Lg = λF∗q for every g ∈ G. In most knownapplications E is a subgroup of F∗q and each Lg is the multiset sum of λ complete systemsof representatives for the cosets of E in F∗q .In [3] it was shown how “playing” with some classic difference sets it is possible toobtain elementary but very useful strong difference families. For instance, it was shownthat if D is a Paley-type difference set in a group G of order 4n − 1 - hence of parameters(4n − 1, 2n − 1, n − 1) - then 2(G \ D) is a (4n − 1, 4n, 4n) difference multiset. Here wewill do something similar using Hadamard PDFs.Proposition 2 If X is a Hadamard (G,K, λ)-PDF, then {2X | X ∈ X } is a (G, 2K, 4λ)-SDF.Cryptogr. Commun. (2019) 11: –5 557 62 559Proof Let m(g) be the multiplicity of g in⊎X∈X( 2X). We have to prove that m(g) = 4λfor all g ∈ G. It is quite evident that if X is a subset of G, then (rX) contains zero exactlyr(r − 1)|X| times and it contains a non-zero element g of G exactly r2λX(g) times whereλX(g) is the multiplicity of g in X. Applying this to our case we getm(0) = 2∑X∈X|X| and m(g) = 4∑X∈XλX(g) ∀g ∈ G \ {0}.Considering that X is a (G,K, λ)-PDF, we have ∑X∈X |X| = |G| and∑X∈X λX(g) = λ.Also, considering that X is a Hadamard PDF, we have |G| = 2λ. Hence we can write:m(0) = 2|G| = 4λ and m(g) = 4λ ∀g ∈ G \ {0}.The assertion follows.We will refer to the strong difference family constructed in the above proposition as theHadamard SDF associated with X .3 The main constructionWe are now ready to state and prove our main result.Theorem 2 Let D be a Hadamard (G,K, λ)-PDF and let R = (H, +, ·) be a ring withidentity of order 2n + 1 admitting a set Y of units such that:Y has size equal to the maximum size Kmax of the blocks of D;every element of (Y ∪ −Y ) is a unit of R.Then there exists a (2λ(2n + 1), n(2K)  {2λ}, 2λ)-PDF in G × H .Proof We get the result by applying Theorem 1 with X the Hadamard (G, 2K, 4λ)-SDFassociated with D. So, if D = {D1, . . . , Dt }, we have X = {X1, . . . , Xt } with Xi = 2Difor i = 1, . . . , t .Take any map f : G −→ Y which is injective on each block Di ∈ D. This is possiblesince the blocks ofD are disjoint by definition and we have |Y | = Kmax ≥ |Di | for every i.Now, for i = 1, . . . , t , consider the subset Bi of G × H defined byBi =⋃d∈Di{d} × {f (d), −f (d)}.As prescribed by Theorem 1 the projection of Bi on G coincides with Xi for i = 1, . . . , t .So we have {B1, . . . , Bt } = ⊎g∈G{g} × Lg where each Lg is a list of 4λ elements of H .Explicitly, these lists are as follows:L0 = {±2f (d) | d ∈ G};Lg = {±f (d) ± f (d ′) | (d, d ′) ∈t⋃i=1Di × Di; d − d ′ = g} for g = 0with all possible choices of the signs. We notice two things: all elements of these listsare units in view of the properties of Y ; each Lg is closed under taking opposites (h and−h have the same multiplicity in Lg for every pair (g, h) ∈ G × H ). Thus we can writeLg = {1, −1} · L′g where each L′g is a list of 2λ units of R.Cryptogr. Commun. (2019) 11: –5 557 62560For every h ∈ H , let us denote by εh the endomorphism of (H, +) which is the multi-plication by h in the ring R. Take a complete set S of representatives for the pairs of thepatterned starter1 of (H,+) and consider the set E = {εs | s ∈ S}. We have:⊎ε∈Eε(Lg) =⊎s∈S{s, −s} · L′g = (H \ {0}) · L′g = 2λ(H \ {0})the last equality being true since every element of L′g is a unit and |L′g| = 2λ. Keeping thesame notation used in Theorem 1 we conclude thatF := {̂εs(Bi) | 1 ≤ i ≤ t; s ∈ S}is a (G × H, G × {0}, n(2K), 2λ)-DF. Now note that we have:⊎s∈Sε̂s(Bi) =⊎s∈S⊎d∈Di{d} × {sf (d),−sf (d)}=⊎d∈Di{d} ×⊎s∈S{s,−s} · f (d) =⊎d∈Di{d} × (H \ {0}) = Di × (H \ {0}).Recalling that the Di’s partition G since D is partitioned, we conclude that the blocks of Fpartition G × (H \ {0}) = (G × H) \ (G × {0}).In order to “complete”F to an ordinary PDF in G×H we need a set of blocks partition-ing G × {0} whose list of differences gives 2λ times (G \ {0}) × {0}. Such a set is triviallygiven by the singleton {G × {0}}. We conclude that{̂εs(Bi) | 1 ≤ i ≤ t; s ∈ S} ∪ {G × {0}}is a (2λ(2n + 1), n(2K)  {2λ}, 2λ)-PDF in G × H .As a corollary we get an infinite class of PDFs applying the above theorem with the useof the Hadamard PDFs of Proposition 1.Corollary 1 IfD is (4u2, 2u2−u, u2−u)-HDS in G and the maximal prime power divisorsof 2n + 1 are all greater than 4u2 + 2u, then there exists a(4u2(2n + 1), [n(4u2 − 2u), 1(4u2), n(4u2 + 2u)], 4u2]-PDF.Proof Let q1, . . . , qt be the maximal prime power divisors of 2n + 1, assume that eachqi is greater than 4u2 + 2u, and let ρi be a primitive root of Fqi . Then the assertion willfollow by applying Theorem 2 withD the Hadamard PDF of Proposition 1, with R the ringFq1 × · · · × Fqt , and with Y = {(ρj1 , . . . , ρjt ) | 1 ≤ j ≤ 2u2 + u}.Applying the above corollary with u = 1, namely using the trivial (4, 1, 0)-HDS, oneobtains a cyclic (8n + 4, [n2, 14, n6], 4)-PDF whenever 2n + 1 is coprime with 15.It is known that there exists a (16, 6, 2)-HDS in every abelian group G of order 16 exceptG = Z16. Then Corollary 1 gives a (400, [1212, 116, 1220], 16)-PDF in G × Z5 × Z5for any G as above. Note, however, that Theorem 2 cannot produce a PDF with the sameparameters in G × Z25 since a set Y of units of Z25 such that (Y ∪ −Y ) ⊂ U(Z25) hassize at most 3.1The patterned starter of an additive group H of odd order is the set of all possible pairs {h,−h} of oppositeelements of H \ {0} (see, e.g., [11]).Cryptogr. Commun. (2019) 11: –5 557 62 561We get another sporadic class of PDFs applying our main theorem with the use of the(32, [22, 6, 22], 16)-PDF given at the end of the introduction.Corollary 2 If the maximal prime power divisors of 2n + 1 are all greater than 44, thenthere exists a(64n + 32, [ 2n4, n12, 132, n44], 32)-PDF.The first value of n for which the above corollary can be applied is 23. In this way onegets a (1504, [464, 2312, 132, 2344], 32)-PDF.Acknowledgements This work has been performed under the auspices of the G.N.S.A.G.A. of the C.N.R.(National Research Council) of Italy.References1. Arasu, K.T., Sehgal, S.: Cyclic difference covers. Aust. J. Commun. 32, 213–223 (2005)2. Buratti, M.: Pairwise balanced designs from finite fields. Discret. Math. 208/209, 103–117 (1999)3. Buratti, M.: Old and new designs via strong difference families. J. Combin. Des. 7, 406–425 (1999)4. Buratti, M., Gionfriddo, L.: Strong difference families over arbitrary graphs. J. Combin. 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Hadamard partitioned difference families and their descendants

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Hadamard partitioned difference families and their descendants

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