Efficient Two-Stage Group Testing Algorithms for DNA Screening

Efficien t Tw o-Stage Group T esting Algorithms for DNA Screening Michael Huber ⋆ Wilhelm-Schic k ard-I nstitute for Co mputer Science Universit y of T uebingen Sand 13, 72076 T uebingen, German y . michael.hu ber@uni-t uebingen.d e Abstract. Group testing algorithms are very useful to ols for DN A library screening. Building on recent w ork by Lev enshtein (2003) and T o n c hev (2008), w e constru ct in th is pap er new infinite classes of com bi- natorial structures, th e existence of which are essen t ial for attaining the minim um num b er of individual tests at the second stage of a tw o-stage disjunctive testing algorithm. 1 In tr o duction With the completion of g enome sequencing pro jects suc h a s the Human Genome Pro ject, efficient screening of DNA clones in very large geno me sequence data- bases has b ecome an impo rtant issue per taining to the study of gene functions. V ery useful to ols for DNA libr a ry screening are gr ou p testing algorithms . The general group testing pro blem (cf. [ 9 ]) can b e basically stated as follows: a large po pulation X of v items that contains a small set of defe ctive (or p ositive ) items shall b e tested in or der to ident ify the defective items efficien tly . F or this, the items are p o oled together for tes ting. The gr oup test repo r ts “yes” if for a subset S ⊆ X one or mor e defective items ha ve b een found, and “no” other wise. Using a n um ber of group tests, the task of determining which items are defective shall be accomplished. V ar ious o b j ectives could be considered for gro up testing, e.g., minimizing the num b er of g roup tests, limiting the num b er of p o ols or po ol sizes, or tolerating a few errors . In what follows, we will fo cus o n the fir s t issue. Of particular practical imp ortance in DNA libr a ry scre e ning are one- o r t wo- stage group testing pro cedur es (cf. [ 18 , p. 371]): “[...] The t ec h nicians who i mplement t he po oling strategies generall y dislike even th e 3-stage strategies that are often u sed. Th us the most commonly u sed strategies for p o oling libraries of clones rely on a fixed bu t reasonably small set on n on-singleton p o ols. The p ools are either tested all at once or in a small num b er of stages (usually at most 2) where the previous stage determines whic h p o ols to test in t h e next stage. The p otentia l p ositiv es are then inferred and confirmed by testing of ind iv idual clones [...].” ⋆ The author gratefull y ackno wledges supp ort of his w ork by the Deutsc he F orsc hungs- gemeinsc haft (DFG) via a Heisen b erg grant (H u954/4) and a Heinz Maier-Leibnitz Prize grant (Hu954/5). 2 Disjunctive testing relies on Bo o lean op erations. It aims to find the set o f de- fective items by reco nstructing its bina r y (0 , 1)-incidence vector x = ( x 1 , . . . , x v ), where x i = 1 if the i th item is defective, a nd x i = 0 otherwise. Lev enshtein [ 20 ] (cf. also [ 25 ]) has employ ed a tw o-stag e disjunctiv e tes ting alg orithm in order to reconstruct the vector x : A t Stage 1, disjunctive tests are conducted whic h are determined by the rows of a binary matrix that is compa r able to a pa rity- chec k matrix of a binary linear co de. After determining what items ar e p ositive, negative or unresolved, individual tests ar e p erfor med a t Stag e 2 in or der to determine which o f the re ma ining unresolved items are po s itive or neg ative. Particularly imp or tant with resp ect to the resear ch ob j ectives in this pap er , Levensh tein derived a combinatorial low er b ound on the minim um num b er of individual t ests at Stage 2. He sho wed that this bound is met with equality if and only if a Steiner t -design exists which has the a dditional pro p er t y that the blo cks hav e tw o siz e s differing by o ne (i.e., k and k + 1). Relying on this result, T onchev [ 25 ] gave a straightforw ard constructio n metho d for such desig ns: Suppo se that D = ( X , B ) is a Steiner t -( v , k , 1) design that con ta ins a Steiner ( t − 1)-( v , k , 1) subdes ign D ′ = ( X, B ′ ), where B ′ ⊆ B . Then, the blocks o f D ′ , each extended with one new p oint x / ∈ X , together with the blo cks of D that do not b e long to D ′ , form a Steiner t -( v + 1 , { k , k + 1 } , 1 ) desig n. Relying on sp ecific balanced incomplete block designs (BIBDs), he constructed tw o infinite classes of such designs: A Steiner 2-( q e + 1 , { q , q + 1 } , 1 ) desig n exis ts for every pr ime power q and every p ositive integer e ≥ 2, and a Steiner 2-(6 a + 4 , { 3 , 4 } , 1) design for every positive integer a , based on reso lv able BIBDs from affine geometr ies and K irkman triple s ystems, r esp ectively . Mo reov er , he constructed tw o infinite classes der ived from cer tain Steiner qua druple sy s tems. In this pap er, we build on the work by Levensh tein and T onchev and con- struct several further infinite classes o f Steiner designs with the desired additional prop erty . Our constructions inv olve, in ter alia, reso lv able BIBDs and cyclically resolv able BIBDs. As a result, we obtain efficient tw o-s tage disjunctiv e group testing algorithms s uited fo r DNA library screening. The pap er is orga niz e d a s follows: Combinatorial to o ls and structur es which are impor tant fo r our further purposes are provided in Sectio n 2 . Section 3 presents a sho r t overview of the previous combinatorial approaches and construc- tions by Levensh tein and T onchev. Section 4 is devoted to our new combinatorial constructions. The pap er concludes in Section 5 . 2 Com binatorial Structures and T o ols Let X b e a set of v ele ments and B a collection of k -subsets of X . The elements of X and B a re called p oints a nd blo cks , resp ectively . An ordered pair D = ( X , B ) is defined to b e a t - ( v , k , λ ) d esign if each t - subset of X is contained in exac tly λ blo cks. F or historical r easons, a t -( v , k , λ ) design with λ = 1 is called a Steiner t -design or a S teiner system . W ell-known examples are Steiner t riple syst ems ( t = 2, k = 3) and S t einer quadruple systems ( t = 3, k = 4). A 2 -design is co mmonly called a b alanc e d inc omplete blo ck design , and de no ted by B IB D( v , k, λ ). It can 3 be e a sily se e n that in a t -( v , k, λ ) design each p oint is contained in the same nu m be r r o f blocks, and for the total n umber b o f blo cks, the parameters of a t -( v , k , λ ) design satisfy the relations bk = v r and r ( k − 1 ) = λ  v − 2 t − 2   k − 2 t − 2  ( v − 1) for t ≥ 2 . Example 1. T a ke as p oint-set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } and a s blo ck-set B = {{ 1 , 2 , 3 } , { 4 , 5 , 6 } , { 7 , 8 , 9 } , { 1 , 4 , 7 } , { 2 , 5 , 8 } , { 3 , 6 , 9 } , { 1 , 5 , 9 } , { 2 , 6 , 7 } , { 3 , 4 , 8 } , { 1 , 6 , 8 } , { 2 , 4 , 9 } , { 3 , 5 , 7 }} . This gives a BIBD(9 , 3 , 1), i.e . the unique affine plane of order 3. It ca n b e constructed as illustr ated in Figure 1 . 1 4 7 2 5 8 3 6 9 Fig. 1. A BIBD(9 , 3 , 1). In this paper, w e primarily fo cus on BIBDs. Let ( X , B ) b e a BIBD( v , k , λ ), and let σ be a p er m uta tion on X . F or a blo ck B = { b 1 , . . . , b k } ∈ B , define B σ : = { b σ 1 , . . . , b σ k } . If B σ : = { B σ : B ∈ B } = B , then σ is called an automorphism of ( X, B ). If there exists an automorphism σ of order v , then the BIBD is called cyclic . In this case, the p oint-set X can b e identified with Z v , the set o f integers mo dulo v , and σ can b e represented by σ : i → i + 1 (mod v ). F or a blo ck B = { b 1 , . . . , b k } in a cyclic BIBD( v, k, λ ), the s et B + i : = { b 1 + i (mo d v ) , . . . , b k + i (mo d v ) } for i ∈ Z v is called a t r anslate of B , and the set of all distinct translates of B is called the orbit containing B . If the length of an orbit is v , then the orbit is sa id to be ful l , o therwise short . A blo ck c hosen 4 arbitrar ily fro m an orbit is called a b ase blo ck (or starter blo ck ). If k divides v , then the or bit containing the blo ck B =  0 , v k , 2 v k , . . . , ( k − 1) v k  is calle d a r e gular short orbit . F or a cyclic BIBD( v , k , 1) to exist, a necessar y condition is v ≡ 1 or k (mod k ( k − 1)). When v ≡ 1 (mod k ( k − 1)) all orbits are full, whereas if v ≡ k (mo d k ( k − 1)) one or bit is the regular short orbit and the remaining o rbits a re full. A BIBD is said to be r esolvable , and denoted by RBIBD( v , k, λ ), if the block- set B ca n be partitioned in to cla sses R 1 , . . . , R r such that every p oint of X is contained in e x actly one blo ck of ea ch cla ss. The clas ses R i are called r esolution (or p ar al lel ) cl asses . A simple e x ample is as follows. Example 2. An RBIBD(9 , 3 , 1). Each row is a reso lution cla ss. R 1 { 1 , 2 , 3 } { 4 , 5 , 6 } { 7 , 8 , 9 } R 2 { 1 , 4 , 7 } { 2 , 5 , 8 } { 3 , 6 , 9 } R 3 { 1 , 5 , 9 } { 2 , 6 , 7 } { 3 , 4 , 8 } R 4 { 1 , 6 , 8 } { 2 , 4 , 9 } { 3 , 5 , 7 } Generally , an RBIB D ( k 2 , k , 1) is equiv alen t to a n affine plane of order k . An RBIBD( v , 3 , 1) is c a lled a Kirkman triple system . Necessar y conditio ns for the existence of an RBIBD( v , k , λ ) are λ ( v − 1 ) ≡ 0 (mo d ( k − 1 )) a nd v ≡ 0 (mo d k ). If R i is a resolution class, define R σ i : = { B σ : B ∈ R i } . An RBIBD is called cyclic al ly r esolvable , and denoted by CRBIBD( v , k , λ ), if it has a non- trivial automorphism σ of or der v that prese rves its resolution {R 1 , . . . R r } , i.e., {R σ 1 , . . . R σ r } = {R 1 , . . . R r } holds. An example is as follows (cf. [ 12 ]). Example 3. A CRBIBD(21 , 3 , 1) is given in T able 1 . The base blocks are { 1 , 4 , 16 } , { 19 , 20 , 3 } , { 1 , 11 , 19 } , and { 0 , 7 , 14 } . There ar e three full o rbits and one r egu- lar short o rbit. Each row is a res olution class. One or bit of reso lution classes is {R 0 , . . . , R 6 } , and a no ther o rbit is {R ′ 0 , R ′ 1 , R ′ 2 } . Mishima and Jimbo [ 21 ] class ified CRBIB D( v , k, 1)s in to three t yp es, ac- cording to their r elation with cyclic quasiframes, cyclic semifra mes, or cy clically resolv able gr oup divisible designs. They can only exist when v ≡ k (mod k ( k − 1)). In a cyclic BIBD( v , k, 1), we can define a m ultiset ∆B : = { b i − b j : i, j = 1 , . . . , k ; i 6 = j } for a base blo ck B = { b 1 , . . . , b k } . Let { B i } i ∈ I , for some index set I , be a ll the base blo cks o f full orbits. If v ≡ 1 (mod k ( k − 1)), then clear ly [ i ∈ I ∆B i = Z v \ { 0 } . The family of base blocks { B i } i ∈ I is then called a (cyc lic) differ enc e family in Z v , denoted by CDF( v , k, 1 ). 5 T able 1. Exa mple of a CRBIBD (21 , 3 , 1). R 0 { 1 , 4 , 16 } { 8 , 11 , 2 } { 15 , 18 , 9 } { 19 , 20 , 3 } { 5 , 6 , 10 } { 12 , 13 , 17 } { 0 , 7 , 14 } R 1 { 2 , 5 , 17 } { 9 , 12 , 3 } { 16 , 19 , 10 } { 20 , 0 , 4 } { 6 , 7 , 11 } { 13 , 14 , 18 } { 1 , 8 , 15 } R 2 { 3 , 6 , 18 } { 10 , 13 , 4 } { 17 , 20 , 11 } { 0 , 1 , 5 } { 7 , 8 , 12 } { 14 , 15 , 19 } { 2 , 9 , 16 } R 3 { 4 , 7 , 19 } { 11 , 14 , 5 } { 18 , 0 , 12 } { 1 , 2 , 6 } { 8 , 9 , 13 } { 15 , 16 , 20 } { 3 , 10 , 17 } R 4 { 5 , 8 , 20 } { 12 , 15 , 6 } { 19 , 1 , 13 } { 2 , 3 , 7 } { 9 , 10 , 14 } { 16 , 17 , 0 } { 4 , 11 , 18 } R 5 { 6 , 9 , 0 } { 13 , 16 , 7 } { 20 , 2 , 14 } { 3 , 4 , 8 } { 10 , 11 , 15 } { 17 , 18 , 1 } { 5 , 12 , 19 } R 6 { 7 , 10 , 1 } { 14 , 17 , 8 } { 0 , 3 , 15 } { 4 , 5 , 9 } { 11 , 12 , 16 } { 18 , 19 , 2 } { 6 , 13 , 20 } R ′ 0 { 1 , 11 , 9 } { 4 , 14 , 12 } { 7 , 17 , 15 } { 10 , 20 , 18 } { 13 , 2 , 0 } { 16 , 5 , 3 } { 19 , 8 , 6 } R ′ 1 { 2 , 12 , 10 } { 5 , 15 , 13 } { 8 , 18 , 16 } { 11 , 0 , 19 } { 14 , 3 , 1 } { 17 , 6 , 4 } { 20 , 9 , 7 } R ′ 2 { 3 , 13 , 11 } { 6 , 16 , 14 } { 9 , 19 , 17 } { 12 , 1 , 20 } { 15 , 4 , 2 } { 18 , 7 , 5 } { 0 , 10 , 8 } Let k b e a n o dd p ositive integer, a nd p ≡ 1 (mo d k ( k − 1)) a prime. A CDF( p, k , 1) is said to b e r adic al , a nd deno ted by RDF( p, k , 1), if each ba se blo ck is a coset of the k -th ro ots of unit y in Z p (cf. [ 3 ]). A link to CRBIBDs has bee n e stablished by Genma, Mishima and Jimbo [ 12 ] as follows. Theorem 1. If ther e is an RDF ( p, k , 1 ) with p a prime and k o dd, t hen ther e exists a CRBIBD ( pk, k, 1) . The notio n of reso lv a bility holds in the same wa y for t -( v , k, λ ) des igns with t ≥ 2. Mor eov er, a Steiner quadruple sys tem 3- ( v , 4 , 1) is ca lled 2-r esolvable if its block-set can b e partitioned in to disjoint Steiner 2-( v , 4 , 1) designs. F or encyclop edic refer ences on combinatorial desig ns, we refer the reader to [ 2 , 8 ]. A comprehensive b o ok on RBIBDs and related designs is [ 11 ]. Highly regular designs are treated in the monogra ph [ 15 ]. A recent surv ey o n v a rious connections b etw een err or-cor recting co des and algebra ic combinatorics is given in [ 16 ]. F or an overview of numerous applications of combinatorial designs in computer and co mm unication s ciences, s ee, e. g., [ 6 , 7 , 17 ]. 3 Com binatorial A pproac hes and Constructions b y Lev ensh tein and T onc hev This section present s a short o verview of r ecent combinatorial approaches and constructions by Levensh tein [ 2 0 ] a nd T onchev [ 2 5 ]. 3.1 Tw o-Stage Disjunctive Group T esting Algorithm Disjunctive group tes ting r elies on Bo olea n o per ations in or der to solve the problem o f r econstructing an unknown binar y vector x of length v using the 6 po o l testing pro cedure [ 9 ]. Particular ly , Levensh tein [ 20 ] (see als o [ 2 5 ]) has employ ed a tw o-s tage disjunctive testing a lgorithm to recons tr uct the vector x = ( x 1 , . . . , x v ): At Stage 1, disjunctive tests are conducted which are deter - mined by the rows of a binary u × v matrix H = ( h i,j ) that is comparable to a pa rity-c heck matrix o f a bina r y linear co de. A syndr ome s = ( s 1 , . . . , s u ) is calculated, where s i is defined by s i = v _ j =1 x j & h i,j , i = 1 , . . . , u, where ∨ and & denote the log ic al op era tions of disjunction a nd co njunction. The sy stem of u logica l equations with v Bo olea n v ariables for reconstructing the vector x = ( x 1 , . . . , x v ) do es not hav e a unique solution in g eneral. After determining wha t items a re p ositive, negative o r unresolved, individual tests are per formed at Stage 2 in order to deter mine whic h of the remaining unr esolved items are p ositive o r negative. 3.2 Minimum Number of Individual T e sts Let X ( v ) be the set of all 2 v subsets of the set X = { 1 , 2 , . . . , v } and X t ( v ) = { x ∈ X ( v ) : | x | = t } . F or a fixed t (1 ≤ t ≤ v ) consider a covering ope r ator F : X t ( v ) → X ( v ) such that x ⊆ F ( x ) for any x ∈ X t ( v ). Define D = { F ( x ) : x ∈ X t ( v ) } . F or any T , 1 ≤ T ≤  v t  , consider the dec r easing contin uous function g t ( T ) = k + k +1 t (1 − α ) w he r e k a nd α ar e uniquely deter mined by the c onditions T  k t  = α  v t  , k ∈ { t, . . . , v } , and 1 − t k +1 < α ≤ 1. Using av eraging and linear progra ming, Levensh tein [ 20 ] prov ed the following inequality: Theorem 2 (Levensh tein, 2003). 1  v t  X x ∈ X t ( v ) | F ( x ) | ≥ g t ( |D| ) , and the b ound is met with e qu ality if and only if D is a Steiner t - ( v , { k, k + 1 } , 1 ) design. One of the main motiv ations for the a bove r esult is to minimize the num b er of individual tests at the second stage of a tw o -stage disjunctive group testing algorithm under the condition that the vectors x a re dis tributed with probabil- ities p | x | (1 − p ) v −| x | where x ∈ X ( v ) denotes the indices of the ones (defective items) in x . The bound ab ov e implies that the expe c ted num b er of items which remain unresolved after applica tion in para llel of u p o ols is not less tha n v v X t =1  v t  p t (1 − p ) v − t 2 − u t − v p. 7 3.3 Kno wn Infinite Classes of Com bi natorial Constructions T onchev [ 25 ] straightforw a rdly gave a non-trivial co nstruction metho d to obtain Steiner desig ns which hav e the additional prop erty that the blo cks hav e tw o s iz es differing by one. Prop ositio n 1 (T onc hev, 2008). Supp ose t hat D = ( X, B ) is a Steiner t - ( v , k , 1) design that c ont ains a Steiner ( t − 1) - ( v , k , 1) su b design D ′ = ( X , B ′ ) , wher e B ′ ⊆ B . Then, the blo cks of D ′ , e ach extende d with one new p oint x / ∈ X , to gether with the blo cks of D that do not b elong to D ′ , form a Steiner t - ( v + 1 , { k , k + 1 } , 1) design. In p articular, if ther e exists an RBIBD ( v , k, 1 ) , then ther e exists a St einer 2 - ( v + 1 , { k , k + 1 } , 1) design. Relying on r esolv able BIBDs from a ffine geo metries and Kirk ma n tr iple sys - tems, T onchev derived from the ab ove re sult the following infinite classes: Theorem 3 (T onc h e v, 2008). Ther e exists • a Steiner 2 - ( q e + 1 , { q , q + 1 } , 1) design for any prime p ower q and any p os- itive inte ger e ≥ 2 , • a Steiner 2 - (6 a + 4 , { 3 , 4 } , 1) design for any p ositive inte ger a . Based on results on 2-r esolv able Steiner quadruple systems by Bak er [ 1 ] & Semako v et a l. [ 22 ] and by T eirlinck [ 23 ], T onchev obtained this wa y also tw o infinite cla sses for t > 2. The third class had a lready b een constructed earlier by T onchev [ 24 ]. Theorem 4 (T onc h e v, 1996 & 2008). Th er e is • a Steiner 3 - (2 2 f + 1 , { 4 , 5 } , 1) design for any p ositive int e ger f ≥ 2 , • a Steiner 3 - (2 · 7 e + 3 , { 4 , 5 } , 1) design for any p ositive int e ger e , • a Steiner 4 - (4 e + 1 , { 5 , 6 } , 1) design for any p ositive inte ger e ≥ 2 . 4 New Infinite Classes of Combina t orial Constr uct ions W e present several constructions o f ne w infinite families of Steiner designs hav- ing the desir ed additional prop erty that the blo cks ha ve t wo sizes differing b y one. Our constructio ns in volve, inter alia , r esolv able BIBDs and cyclica lly reso lv- able BIBDs. As a res ult, we obtain efficient tw o-s tage dis junctiv e group testing algorithms suited for DNA library screening. 4.1 CRBIBD-Constructions W e obtain the following res ult: Theorem 5. L et p b e a prime. Then ther e exists a S teiner 2 - ( pk + 1 , { k, k + 1 } , 1 ) design for the fol lowing c ases: 8 (1) ( k , p ) = (3 , 6 a + 1) for any p ositive int e ger a , (2) ( k , p ) = (4 , 12 a + 1) f or any o dd p ositive inte ger a , (3a) ( k , p ) = (5 , 20 a + 1) for any p ositive inte ger a such t hat p < 10 3 , and fur- thermor e (3b) ( k , p ) = (5 , 20 a + 1) for any p ositive inte ger a satisfying the c ondition state d in (ii) in the pr o of, (4) ( k , p ) = (7 , 42 a + 1) for any p ositive inte ger a satisfying the c ondition state d in (iii) in the pr o of, (5) ( k , p ) = (9 , p ) for the values of p ≡ 1 (mod 72) < 10 4 given in T able 2 . Mor e over, t her e exist s a St einer 2 - ( q k + 1 , { k , k + 1 } , 1 ) design for the fol- lowing c ases: (6) ( k , q ) for k = 3 , 5 , 7 , or 9 , and q is a pr o duct of primes of the form p ≡ 1 (mo d k ( k − 1)) as in t he c ases ab ove, (7) ( k , q ) = (4 , q ) and q is a pr o duct of primes of t he form p = 12 a + 1 with a o dd. Pr o of. The constr uctions are bas ed on the existence of a CRBIBD( pk , k, 1) in conjunction with P rop osition 1 . W e firs t assume that k is o dd. Then the follow- ing infinite ((i)-(iii)) and finite ((iv)) families of r adical difference families exist (cf. [ 3 ] and the references therein; [ 8 ]): (i) An RDF( p, 3 , 1 ) exists for all primes p ≡ 1 (mo d 6). (ii) Let p = 2 0 a + 1 b e a prime, let 2 e be the largest power of 2 dividing a a nd let ε be a 5-th pr imitiv e roo t of unity in Z p . Then an RDF( p, 5 , 1) exists if and o nly if ε + 1 is not a 2 e +1 -th p ow er in Z p , or equiv alently (11 + 5 √ 5) / 2 is not a 2 e +1 -th p ow er in Z p . (iii) Let p = 42 a + 1 be a prime and let ε b e a 7-th primitive r o ot of unity in Z p . Then an RDF ( p, 7 , 1) exists if and only if there exists an integer f such that 3 f divides a and ε + 1 , ε 2 + ε + 1 , ε 2 + ε +1 ε +1 are 3 f -th p ow ers but not 3 f +1 -th powers in Z p . (iv) An RDF( p, 9 , 1) exists fo r all primes p < 10 4 display ed in T able 2 . Theorem 1 yields the re s pec tive CRBIBD( pk, k, 1)s. Mor eov er, in [ 12 ] a re- cursive construction is given that implies the existence of a CRBIBD( k q , k, 1 ) whenever q is a pr o duct of primes of the form p ≡ 1 (mod k ( k − 1)). In addition, a CRBIBD(5 p, 5 , 1 ) has b een shown [ 4 ] to exist for any prime p ≡ 1 (mod 20) < 10 3 . W e now consider the cas e when k is even: In [ 19 ], a CRBIBD(4 p, 4 , 1) is constructed for any pr ime p = 20 a + 1, where a is a n o dd p ositive integer. F urthermore, v ia the a bove recur s ive co nstruction, a CRBIBD(4 q , 4 , 1) exists whenever q is a pro duct of primes of the form p = 12 a + 1 a nd a is odd. The result fo llows. ⊓ ⊔ Example 4. V alues of p for which an RDF( p, k , 1 ) exists with k = 5 , p < 10 3 , and k = 7 or 9, p < 10 4 are displayed in T able 2 (cf. [ 8 ]). F or example, if we take an RDF(4 1 , 5 , 1), then we obtain a Steiner 2-(20 6 , { 5 , 6 } , 1) design. If we take a n RDF(61 , 5 , 1), then we obtain a Steiner 2-(3 06 , { 5 , 6 } , 1) design. 9 T able 2. Exis tence of an RDF( p, k , 1) with k = 5, p < 10 3 , and k = 7 or 9, p < 10 4 . k = 5 41 61 241 281 401 421 601 641 661 701 761 821 881 k = 7 337 421 463 883 1723 3067 3319 3823 3907 4621 4957 5167 5419 5881 6133 8233 8527 8821 9619 9787 9829 k = 9 73 1153 1873 2017 6481 7489 7561 W e remark t hat further parameters are given in [ 3 ] for R DF( p, k, 1)s with k = 7 or 9 and 10 4 ≤ p < 10 5 . 4.2 RBIBD-Constructions W e establish the following r e s ult: Theorem 6. L et v b e a p ositive inte ger. Then t her e exists a Steiner 2 - ( v + 1 , { k , k + 1 } , 1) design fo r the fol lowing c ases: (1) ( k , v ) = (3 , 6 a + 3) for any p ositive inte ger a , (2) ( k , v ) = (4 , 12 a + 4) for any p ositive int e ger a , (3) ( k , v ) = (5 , 20 a + 5) for any p ositive int e ger a with the p ossible ex c eptions given in T able 3 , (4) ( k , v ) = (8 , 56 a + 8) for any p ositive int e ger a with the p ossible ex c eptions given in T able 3 . Pr o of. The constructions ar e based on the existence of an RBIBD( v , k , 1) in con- junction with Pro p os ition 1 . The following infinite series of r e solv able ba lanced incomplete blo ck designs a re known (cf. [ 13 , 8 ] and the refere nces there in): (i) When k = 3 and 4 , resp ectively , an RBIBD( v , k , 1) exists fo r all p os itiv e int egers v ≡ k (mo d k ( k − 1 )). (ii) An RBIBD( v , 5 , 1) exists for a ll po s itiv e in teger s v ≡ 5 (mo d 20) with the po ssible e x ceptions g iven in T able 3 . (iii) An RBIBD( v , 8 , 1) exists for a ll po s itiv e in teger s v ≡ 8 (mo d 56) with the po ssible e x ceptions g iven in T able 3 . This prov es the theorem. ⊓ ⊔ W e remark that Case (1) ha s a lr eady b een cov er ed in Theorem 3 . 10 Example 5. Cho os ing for example an RBIBD(65 , 5 , 1), w e g et a Steiner 2-(66 , { 5 , 6 } , 1) design. If w e choos e an RBIBD(105 , 5 , 1), then we obtain a Steiner 2-(106 , { 5 , 6 } , 1) des ign. T able 3. Possible e xceptions: An RBIBD( v , k , 1) with k = 5 or 8 is not known to exist for the following v alues of v ≡ k (mo d k ( k − 1)). k = 5 45 345 465 645 k = 8 176 624 736 1128 1240 1296 1408 1464 1520 1576 1744 2136 2416 2640 2920 2976 3256 3312 3424 3760 3872 4264 4432 5216 5720 5776 6224 6280 6448 6896 6952 7008 7456 7512 7792 7848 8016 9752 10200 10704 10760 10928 11040 11152 11376 11656 11712 11824 11936 12216 12328 12496 12552 12720 12832 12888 13000 13280 13616 13840 13896 14008 14176 14232 21904 24480 Theorem 7. If v and k ar e b oth p owers of the same prime, then a S teiner 2 - ( v + 1 , { k , k + 1 } , 1) ex ist s if and only if ( v − 1 ) ≡ 0 (mod ( k − 1 )) and v ≡ 0 (mo d k ) . Pr o of. It has b een shown in [ 14 ] tha t, for v and k b oth powers of the same prime, the necessa ry conditions for the existence of an RBIBD( v , k , λ ) are sufficient. Hence, the result follows via Prop ositio n 1 when considering an RBIBD( v , k , 1).  4.3 3-Design-Co nstructions Based on further results on 2- resolv able Steiner quadruple systems descr ibe d in T eirlinck [ 23 ], w e obtain this way also tw o infinite classes for t > 2. Theorem 8. Ther e exists • a Steiner 3 - (2 · 3 1 e + 3 , { 4 , 5 } , 1) design for any p ositive inte ger e , • a Steiner 3 - (2 · 1 27 e + 3 , { 4 , 5 } , 1) design for any p ositive int e ger e . 11 5 Conclusion Efficient tw o-s ta ge gro up testing algo rithms that ar e par ticular s uited fo r DNA library scre ening hav e be en inv estigated in this pap er. The main fo cus has b een on novel combinatorial constr uctions in or der to minimize the n um be r of individ- ual tests at the second stage of a t wo-stage dis junctive testing algo r ithm. Several infinite c lasses of such combinatorial structur es have b een obtained. References 1. R. D. Baker, “P artitioning the planes of AG 2 m (2) into 2-designs”, Discr ete Math. , vol . 15, pp. 205–211, 1976. 2. Th. Beth, D. Jungnickel and H . Lenz, Design The ory , vol. I and I I, Encyclop edia of Math. and Its A pplications, vo l. 69/78, Cambridge Univ. Press, Cambridge, 1999. 3. M. Buratti, “On simple radical d ifference families”, J. Combin. Designs , vol. 3, pp. 161–16 8, 1995. 4. M. Buratti, “On resolv able difference families”, Designs, Co des and Crypto gr aphy , vol . 11, pp. 11–23, 1997. 5. K. Chen, R. W ei and L. Zhu, “Existence of ( q , 7 , 1) difference families with q a prime pow er”, J. Combi n. Designs , vol. 10, pp. 126–138, 2002. 6. C. J. Colb ourn and P . C. v an Oorschot, “App lications of com b in atorial d esigns in computer science”, ACM Comp. Surveys , vo l. 21, pp. 223–250, 1989. 7. C. J. Colbourn , J. H . Dinitz and D. R. Stinson, “App lications of com b inatorial designs to communications, cryptography , and net working”, in: Surveys in Com - binatorics , ed . by J. D. Lamb and D . A. Preece, London Math. Soc. Lectu re N ote Series, vo l. 267, Cam bridge Univ. Press, Cam b ridge, pp. 37–100, 1999. 8. C. J. Colbourn and J. H. Dinitz (eds.), Handb o ok of Combinatorial Designs , 2nd ed., CRC Press, Bo ca Raton, 2006. 9. D.-Z. D u and F. K. Hw ang, Combinatorial G r oup Testing and i ts Appli c ations , 2nd ed., W orld Scientific, S in gapore, 1999. 10. D.-Z. Du and F. K. Hwa ng, Po oling Designs and Nonadaptive Gr oup Testing: Imp ortant To ols for DNA Se quencing , W orld Scientific, Singap ore, 2006. 11. S. F urino, Y. Miao and J. Y in, F r ames and Resolvable Designs , CRC Press, Boca Raton, 1996. 12. M. Genma, M. Mishima and M. Jimbo, “Cyclic resolv ability of cy clic Steiner 2- designs”, J. C ombin. Designs , v ol. 5, p p. 177–1 87, 1997. 13. M. Greig and J. Ab el, “Resolv able balanced incomplete blo ck designs with b lock size 8”, Designs, Co des and Crypto gr aphy , vol. 11, pp. 123–140 , 1997. 14. M. Greig, “Log tables and block designs”, Bul l. I nst. Combin. Appl. , vol . 48, pp. 66– 72, 2006. 15. M. Hub er, Flag-tr ansitive Steiner Designs , Birkh¨ auser, Basel, Berlin, Boston, 2009. 16. M. Hub er, “Coding theory and algebraic combinatori cs”, in Sele cte d T opics in Information and Co ding The ory , ed. by I. W oungang et al., W orld Scientific, Sin- gapore, pp. 121–158, 2010. 17. M. Hub er, Combinatorial Designs for Authentic ation and Se cr e cy Co des , F ounda- tions an d T rends r in Comm unications and In formation Theory , No w Pu b lishers, Boston, Delft, 2010. 12 18. E. Knill, “Low er b ounds for identi fying subset members with subset queries”, in: Pr o c. 6th A nnual A CM-SIAM Symp osium on Discr ete A lgorithms (SOD A) 1995 , pp. 369–37 7, 1995. 19. C. Lam and Y. Miao, “On cyclically resolv able cyclic Steiner 2-designs”, J. Combi n. The ory, Series A , vol. 85, pp. 194–2 07, 1999. 20. V. I . Levensh tein, “A universal bou n d for a co vering in regular p osets and its application to p ool testing”, Discr ete Math. , vo l. 266, pp. 293–30 9, 2003. 21. M. Mishima and M. Jimbo, “Some typ es of cyclically resolv able cyclic Steiner 2-designs”, Congr. Numer. , vol . 123, pp. 193–203 , 1997. 22. N. V. Semako v , V. A. Zinoviev and G. V. Zaitsev, “Interrelation of Preparata and Hamming cod es and extension of Hamming co des to new double-error-correcting codes”, in: Pr o c. 2nd I nternational Symp osium on Information The ory 1971 , Academiai Kiado, Budap est, pp. 257–263, 1973. 23. L. T eirlinc k, “Some new 2- resolv able S teiner qu ad ru ple systems”, Designs, Co des and C rypto gr aphy , vol. 4, pp . 5–10, 1994. 24. V. D . T onchev, “A class of Steiner 4-wise balanced designs derived from Preparata codes”, J. Combin. Designs , vol. 3, pp . 203–204, 1996. 25. V. D. T onchev, “St einer sy stems for tw o-stage disjunctive testing”, J. Combin. Optim. , vol . 15, pp. 1–6, 2008.