On the periods of generalized Fibonacci recurrences

On the P erio ds of Generalized Fib onacci Recurrences 1 Ric hard P . Bren t Computer Sciences Lab oratory Australian National Univ ersit y Can b e rra, A CT 0200 TR-CS-92-03 Marc h 19 92, revised Marc h 19 93 Abstract W e give a simple condition for a linear recurrence ( mo d 2 w ) of deg r ee r to hav e the maximal p ossible p erio d 2 w − 1 (2 r − 1). It follows that the p erio d is ma x imal in the cases of in terest for pseudo-random n um b er generation, i.e. for 3-term linear recurrenc e s defined by trinomials which are primitive (mod 2 ) and of degree r > 2. W e conside r the enumera- tion of certain exceptional po lynomials which do not give maximal p erio d, and list all s uch po lynomials of deg r ee less than 15. 1 In tro duction The Fib onacci num b ers satisfy a linear r ecurrence F n = F n − 1 + F n − 2 . Gener alize d Fib onac ci r ecurrences of th e f orm x n = ± x n − s ± x n − r mo d 2 w (1) are of inte rest b ecause they are often used to generate pseudo-random num b ers [1, 5, 6, 11 , 13, 17]. W e assume th r oughout that x 0 , . . . , x r − 1 are give n and not all even, and w > 0 is a fixed exp onent. Usu ally w is close to the wordlength of the (b in ary) computer used. Apart from computational con v enience, there is n o reason to restrict atten tion to 3-term recurrences of the sp ecial form (1). Thus, we consid er a general linear recurrence q 0 x n + q 1 x n +1 + · · · + q r x n + r = 0 mo d 2 w (2) defined by a p olynomial Q ( t ) = q 0 + q 1 t + ... + q r t r (3) of degree r > 0. W e assum e throu gh ou t that q 0 and q r are od d. q 0 o dd imp lies that the sequence ( x n ) is r ev ersible, i.e. x n is uniquely defined (mo d 2 w ) by x n +1 , . . . , x n + r . Thus, ( x n ) is pu r ely p erio d ic [19]. In the follo w ing we often w ork in a rin g Z m [ t ] /Q ( t ) of p olynomials (mo d Q ) whose co efficien ts are regarded as element s of Z m (the rin g of in tegers mo d m ). F or relations A = B in Z m [ t ] /Q ( t ) w e use th e notation A = B mo d ( m, Q ) . 1 1991 Mathematics Subje ct Classific ation. Primary 11Y55, 12E05, 05A15; Secondary 11-04, 11T06, 11T55, 12-04, 12E10, 65C10, 68R05. Key wor ds and phr ases. Fibon acci sequence, generalized Fib onacci sequence, irreducible trinomial, linear recurrence, maximal p erio d, p erio dic intege r sequen ce, p rimitive trinomial, p seudo-random numbers. Cop yright c  1992–2 010, R. P . Brent. rpb133tr typeset using L A T E X It may b e sh o w n by indu ction on n that if a n, 0 , . . . , a n,r − 1 are d efined by t n = r − 1 X j =0 a n,j t j mo d (2 w , Q ( t )) (4) then x n = r − 1 X j =0 a n,j x j mo d 2 w . (5) Also, the generating function G ( t ) = X n ≥ 0 x n t n (6) is giv en b y G ( t ) = P ( t ) ˜ Q ( t ) mo d 2 w , (7) where P ( t ) = r − 1 X k =0   k X j =0 q r + j − k x j   t k is a p olynomial of degree less than r , and ˜ Q ( t ) = t r Q (1 /t ) = q 0 t r + q 1 t r − 1 + ... + q r is the r everse of Q . In th e literature, ˜ Q ( t ) is sometimes called the char acteristic p olyno mial [4] or the asso ciate d p olynomial [19] of the sequence. The u s e of generating fu nctions is conv enient and h as b een adopted by man y earlier authors (e.g. S c h ur [15]). W ard [19] do es not explicitly use generating fun ctions, bu t h is p olynomial U is the same as ou r ˜ Q , and man y of h is results could b e obtained via generating functions. Let ρ w b e the p erio d of t under multiplica tion mo d (2 w , Q ( t )), i.e. ρ w is the least p ositiv e in teger ρ s u c h that t ρ = 1 mo d (2 w , Q ( t )) . In the literature, ρ w is sometimes called the princip al p erio d [19] of the linear recurrence, some- times simply th e p erio d [4]. F or br evity we defin e λ = ρ 1 . An irreducible p olynomial in Z 2 [ t ] is a f actor of t 2 r − t (see e.g. [18]), so λ | 2 r − 1. W e sa y that Q ( t ) is primitive (mo d 2) if λ = 2 r − 1. Note that primitivit y is a stronger condition than irreducibilit y 2 , i .e. Q ( t ) pr imitiv e implies that Q ( t ) is irreducible, but th e con verse is not generally tru e unless 2 r − 1 is prime 3 . T ables of ir r educible and primitive trinomials are a v ailable [4, 10, 14, 16, 20, 22, 23, 24 , 25]. In th e follo wing we usually assume th at Q ( t ) is irr ed ucible. Ou r assumption that q 0 and q r are o dd excludes the trivial case Q ( t ) = t , and implies th at ˜ Q ( t ) is irreducible (or primitiv e) of degree r iff the same is tr ue of Q ( t ). W e are in terested in the p erio d p w of the s equence ( x n ), i.e. the minimal p ositiv e p such that x n + p = x n (8) for all sufficient ly large n . In fact, b ecause of the rev ersibilit y of the sequence, (8) should hold for all n ≥ 0. The p erio d is sometimes called the char acteristic numb er of the sequen ce [19]. 2 F or brev ity w e usually omit th e “(mo d 2)” when sa ying that a p olynomial is irreducible or primitive. Thus “ Q ( t ) is irreducible (resp. primitive)” means that Q ( t ) mod 2 is irreducible (resp. primitive) in Z 2 [ t ]. 3 F or example, the p olynomial 1 + t + t 2 + t 4 + t 6 is irreducible, but n ot primitive, since it has λ = 21 < 2 6 − 1. 2 In general the p erio d dep end s on the initial v alues x 0 , . . . , x r − 1 , but under our assumptions the p erio d dep ends only on Q ( t ), in fact p w = ρ w (see Lemma 2). It is kn own [7 , 12, 19] that p w ≤ 2 w − 1 λ with equalit y holding for all w > 0 iff it h olds for w = 3. The main aim of this p ap er is to giv e a simple necessary and suffi cien t cond ition for p w = 2 w − 1 λ. (9) The resu lt is stated in T heorem 2 in terms of a simple condition which w e call “Condition S” (see S ection 2). In Theorem 3 we deduce that th e p erio d is maximal if Q ( t ) is a primitiv e trinomial of degree greater than 2. Th us, in cases of p r actical in terest for pseud o-random n umber generation 4 , it is only necessary to v erify that Q ( t ) is pr imitiv e. This is particularly easy if 2 r − 1 is a Mersen ne prime, b ecause then a necessary and suffi cien t condition is t 2 r = t mo d (2 , Q ( t )) . The basic results on linear recurrences m o dulo m w ere obtained man y ye ars ago – see f or example W ard [19 ]. Ho w ev er, our main r esults (Th eorems 2 and 3) and the statemen t of “Con- dition S” (Section 2) app ear to b e new. 2 A Cond ition for M aximal P erio d The follo wing Lemma is a sp ecial case of Hensel’s Lemma [7, 8, 21] and ma y b e pr o v ed using an application of Newton’s metho d for recipro cals [9]. Lemma 1 Supp ose that P ( t ) mo d 2 is invertible in Z 2 [ t ] /Q ( t ) . Then, for al l w ≥ 1 , P ( t ) mo d 2 w is invertible in Z 2 w [ t ] /Q ( t ) . W e n o w giv e a sufficient condition for the p erio ds p w and ρ w to b e the same. Lemma 2 If Q ( t ) is irr e ducib le of de gr e e r and at le ast one of x 0 , . . . , x r − 1 is o dd, then p w = ρ w . Pro of F or br evit y we wr ite p = p w and ρ = ρ w . F rom (6), G ( t ) = R ( t ) 1 − t p mo d 2 w , where R ( t ) h as degree less than p . Th us, fr om (7), R ( t ) ˜ Q ( t ) = (1 − t p ) P ( t ) mo d 2 w . (10) No w P ( t ) mo d 2 h as degree less than r , bu t is not iden tically zero. Sin ce ˜ Q ( t ) mo d 2 is irredu cible of degree r , application of the extended Euclidean algorithm [7 ] to P ( t ) mo d 2 and ˜ Q ( t ) mo d 2 constructs the in v erse of P ( t ) mo d 2 in Z 2 [ t ] / ˜ Q ( t ). Thus, Lemma 1 sho ws that P ( t ) mo d 2 w is in v ertible in Z 2 w [ t ] / ˜ Q ( t ). It follo ws from (10) that t p = 1 mod (2 w , ˜ Q ( t )) , 4 A word of caut ion is appropriate. Even when the p eriod p w satisfies (9), it is not desirable to use a full cycle of p w num b ers in applications requiring indep end ent pseudo-random numbers. This is b ecause only the most significan t bit has the full p erio d . I f the bits are num b ered from 1 (least significan t) to w (most significan t), then bit k has p erio d p k . 3 and ρ | p . How ev er, from (4) and (5), p | ρ . Thus p = ρ . ✷ As an example, consider Q ( t ) = 1 − t + t 2 . W e ha v e t 3 = 1 mo d (2 , Q ( t )), t 3 = − 1 mo d Q ( t ), and t 6 = 1 mo d Q ( t ), so ρ w =  3 , if w = 1; 6 , if w > 1. (11) It is easy to v erify that (11) give s the p erio d p w of the corresp onding recurrence x n = x n − 1 − x n − 2 mo d 2 w pro vided x 0 and x 1 are not b oth ev en. The assum ption of irredu cibilit y in Lemma 2 is s ignifican t. F or example 5 , consid er Q ( t ) = t 2 − 1 and w = 1, with initial v alues x 0 = x 1 = 1. Th e recurrence is x n = x n − 2 mo d 2, so p 1 = 1, but ρ 1 = 2. Here P ( t ) = 1 + t is a d ivisor of ˜ Q ( t ) = 1 − t 2 . W e no w defin e a condition w hic h must b e s atisfied by Q ( ± t ) if the p er io d p w of the s equence ( x n ) is less than 2 w − 1 λ (see Theorem 2 for details). F or giv en Q ( t ) the condition can b e chec k ed in O ( r 2 ) op erations 6 . This is muc h faster than the metho d suggested by Knuth [7] or Marsaglia and Tsay [12], wh ic h inv olv es forming high p o w ers of r × r matrices (mo d 8). Condition S Let Q ( t ) = P r j =0 q j t j b e a p olynomial of degree r . W e sa y that Q ( t ) s atisfies Condition S if Q ( t ) 2 + Q ( − t ) 2 = 2 q r Q ( t 2 ) mo d 8 . Lemma 3 giv es an equiv alent condition 7 whic h is more conv enien t for computational pur- p oses. Th e pro of is s tr aigh tforward, so is omitted. Lemma 3 A p olynomial Q ( t ) of de g r e e r satisfies Condition S iff X j + k =2 m 0 ≤ j 1 corresp onds to Q ( − t ) satisfying Cond ition S, wh ile W ard’s case ( T = 1, K ( x ) = 1 mo d 2) corresp onds to Q ( t ) satisfying Cond ition S. How ev er, W ard’s exp osition is complicated by consideration of o dd prime p ow er mo d u li (see for example his Theorem 13.1), so we giv e an ind ep endent pro of. Theorem 2 L et Q ( t ) b e irr e ducible and define a line ar r e curr enc e by (2), with at le ast one of x 0 , . . . , x r − 1 o dd. Then the se quenc e ( x n ) has p erio d p w ≤ 2 w − 2 λ for al l w ≥ 2 i f Q ( − t ) satisfies Condition S, p w ≤ 2 w − 2 λ 6 for al l w ≥ 3 i f Q ( t ) satisfies Condition S, and p w = 2 w − 1 λ for al l w ≥ 1 i ff neither Q ( t ) nor Q ( − t ) satisfies Condition S. Pro of F rom Lemma 2, p w = ρ w is the ord er of t mo d (2 w , Q ( t )). If Q ( − t ) satisfies Condition S then, from Th eorem 1, t λ = 1 mod (4 , Q ( t )) . Using (14), it follo ws by ind uction on w that t 2 w − 2 λ = 1 mo d (2 w , Q ( t )) for all w ≥ 2. This prov es the firs t part of the Theorem. T he second p art is similar, so it only remains to pr o ve the third part. Supp ose that ρ w = 2 w − 1 λ f or all w > 0. In particular, for w = 3 we ha v e p erio d ρ 3 = 4 λ . Th us t 2 λ 6 = 1 mo d (8 , Q ( t )) and, from (16), t λ 6 = ± 1 mo d (4 , Q ( t )) . (29) F rom T h eorem 1, neither Q ( t ) nor Q ( − t ) can satisfy C on d ition S, or w e would obtain a con tra- diction to (29). Con v ersely , if n either Q ( t ) or Q ( − t ) satisfies Condition S , then we show by in duction on w that t 2 w − 1 λ = 1 + 2 w R w mo d Q ( t ) , (30) where R w 6 = 0 mo d (2 , Q ( t )) , (31) for all w ≥ 1. Certainly t λ = 1 mo d (2 , Q ( t )) but, from T h eorem 1, t λ 6 = 1 mod (4 , Q ( t )) , so (30) and (31) hold for w = 1. Defining R w = R w − 1 (1 + 2 w − 2 R w − 1 ) (32 ) for w ≥ 2, we see that (30) holds for all w ≥ 1. I t remains to pro v e (31) for w > 1. F or w = 2, (31) follo ws f r om Theorem 1 an d (16), b ecause t λ 6 = ± 1 mo d (4 , Q ( t )) implies t 2 λ 6 = 1 mo d (8 , Q ( t )). F or w > 2, (31) follo ws by indu ction fr om (32), since 2 w − 2 is ev en. It follo ws that ρ w = 2 w − 1 λ for all w ≥ 1. ✷ 7 3 Primitiv e T rinomial s In this sectio n we consider a case of in terest b ecause of its applications to pseud o-rand om num b er generation: Q ( t ) = q 0 + q s t s + q r t r is a trinomial ( r > s > 0). Theorem 3 sho ws th at the p erio d is alwa ys maximal in cases of practical interest. The cond ition r > 2 is necessary , as the example Q ( t ) = 1 − t + t 2 of Section 2 sho ws. Theorem 3 L et Q ( t ) b e a primitive trinomial of de g r e e r > 2 . Then the se quenc e ( x n ) define d by (2) (with at le ast one of x 0 , . . . , x r − 1 o dd) has p erio d p w = 2 w − 1 (2 r − 1) . Pro of F rom T heorem 2 it is su fficien t to sho w that Q ( t ) do es not satisfy C ondition S . (Since Q ( − t ) is also a trin omial, the s ame argum en t shows that Q ( − t ) d o es not satisfy Cond ition S.) Supp ose, by wa y of contradict ion, that Q ( t ) satisfies Cond ition S. W e use the formulatio n of Condition S giv en in Lemma 3. S ince Q ( t ) is irreducible, q 0 = q s = q r = 1 mo d 2. If s is even, sa y s = 2 m , th en X j + k =2 m 0 ≤ j 2, so 3 s < 2 r − 1, and Q ( t ) can not b e primitiv e. T h is con tradiction completes the pro of. ✷ A minor m o dification of the p ro of of Th eorem 3 giv es: Theorem 4 L et Q ( t ) = q 0 + q s t s + q r t r b e an irr e ducible trinomial of de gr e e r 6 = 2 s . Then the se quenc e ( x n ) define d by (2) (with at le ast one of x 0 , . . . , x r − 1 o dd) has p erio d p w = 2 w − 1 λ . As mentio ned ab ov e, it is easy to find p rimitiv e tr in omials of very high degree r if 2 r − 1 is a Mersenne prime. Zierler [24] giv es examples with r ≤ 9689, and we foun d t wo examp les with higher degree: t 19937 + t 9842 + 1 and t 23209 + t 9739 + 1. These and other examples with r ≤ 44497 w ere f ound indep en d en tly b y Kurita and Matsumoto [10]. S uc h p r imitiv e trinomials provide the basis for f ast ran d om n umber generators w ith extremely long p erio ds and go o d statistica l prop erties [3]. 4 Exceptional P olynomials W e sa y that a p olynomial Q ( t ) of degree r > 1 is exc eptional if conditions 1–3 hold and is a c andidate if conditions 2–3 hold – 1. Q ( t ) mo d 2 is primitiv e. 2. Q ( t ) has co efficients q j ∈ { 0 , − 1 , +1 } , and q 0 = q r = 1. 8 3. Q ( t ) satisfies Condition S. By T h eorem 2, if Q ( t ) is exceptional then Q ( t ) and Q ( − t ) d efine simp le linear recurrences (mo d 2 w ) which hav e less than the maximal p erio d for w > 2. Only the coefficien ts of Q ( t ) mo d 4 are relev an t to Condition S. If condition 2 is relaxed to allo w coefficients equal to 2 then, by Lemma 3, th er e is one suc h Q ( t ) corresp onding to eac h primitive p olynomial in Z 2 [ t ]. With condition 2 as stated the n u m b er of these Q ( t ) is considerably reduced. It is interesting to consider strengthening condition 2 by askin g for certain p atterns in the signs of the coefficients. F or example, we migh t ask for p olynomials Q ( t ) with all co efficien ts q j ∈ { 0 , 1 } , or for all co efficients of ± Q ( − t ) to b e in { 0 , 1 } . Th er e are cand idates satisfying these conditions, but we ha v e n ot found any wh ic h are also exceptional, apart from the trivial Q ( t ) = 1 − t + t 2 . It is p ossible for an exceptional p olynomial to hav e ( − 1) j q j ≥ 0 for 0 ≤ j < r . The only example f or 2 < r ≤ 44 is Q ( t ) = 1 − t + t 2 − t 5 + t 6 + t 8 − t 9 + t 10 + t 12 − t 13 + t 16 + t 18 + t 21 . Observe that Q ( − t ) defines a linear recurrence with nonnegativ e co efficien ts x n +21 = x n + x n +1 + x n +2 + x n +5 + x n +6 + x n +8 + x n +9 + x n +10 + x n +12 + x n +13 + x n +16 + x n +18 whic h has p erio d p 2 = p 1 = 2 21 − 1 when considered mo d 2 or mo d 4. In T able 1 we list the exceptional p olynomials Q ( t ) of d egree r ≤ 14. If Q ( t ) is exceptional then so is ˜ Q ( t ). Thus, w e only list one of these in T able 1. The num b er ν ( r ) of exceptional Q ( t ) (coun ting only one of Q ( t ) , ˜ Q ( t )) is giv en in T able 2. The term “exceptional” is justified as ν ( r ) app ears to b e a m uch m ore slo wly growing fun ction of r than the num b er [4 ] λ 2 ( r ) = ϕ (2 r − 1) / r of primitive p olynomials of degree r in Z 2 [ t ] (where ϕ is Euler’s totien t-function) or the total n umber of p olynomials of degree r w ith co efficient s in { 0 , − 1 , +1 } . Heuristic argum en ts su ggest that the n umber κ ( r ) of candidates should gro w lik e (3 / 2) r and that ν ( r ) should gro w lik e (3 / 4) r λ 2 ( r ). T he argument s are as follo ws – There are 2 r − 1 p olynomials ¯ Q ( t ) of degree r with co efficients in { 0 , 1 } , s atisfying ¯ q 0 = ¯ q r = 1. Randomly select suc h a ¯ Q ( t ), and compu te ǫ 0 , ǫ 1 , . . . , ǫ r from X j + k =2 m 0 ≤ j 2. Ho w ev er, the argum en t d o es app ear to p redict the correct order of m agnitud e of κ ( r ). 9 The pr obabilit y that a r andomly chosen ¯ Q ( t ) with ¯ q 0 = ¯ q r = 1 is primitive is just λ 2 ( r ) / 2 r − 1 . If there is the same p robabilit y th at a randomly c h osen candidate is primitiv e, then the num b er of primitive candid ates s hould b e (3 / 4) r − 1 λ 2 ( r ), and ν ( r ) should b e half this num b er. In T able 2 we giv e ¯ ν ( r ) = ν ( r ) (3 / 4) r λ 2 ( r ) ; the numerical evid en ce suggests that ¯ ν ( r ) con v erges to a p ositiv e constant ¯ ν ( ∞ ) as r → ∞ . Ho wev er, ¯ ν ( ∞ ) is less than th e v alue 2/3 predicted b y the heuristic argument. Our b est estimate (obtained from a separate computation wh ic h giv es faster con v ergence) is ¯ ν ( ∞ ) = 0 . 45882 ± 0 . 00002 The computation of T able 2 to ok 166 hours on a V axStatio n 3100. W e outline the metho d used. It is easy to chec k if a candidate p olynomial is exceptional [7]. A straightfo rward metho d of en umerating all cand idate p olynomials of degree r is to asso ciate a p olynomial Q ( t ) such that q 0 = q r = 1 with an ( r − 1)-bit binary num b er N = b 1 · · · b r − 1 , where b j = q j mo d 2. F or eac h suc h N , compute ǫ 0 , . . . , ǫ r from (12). No w (13) d efines q 0 , . . . , q r mo d 4. If there is an index m suc h th at ǫ m = 1 mo d 2 but q m = 0 m o d 2, th en (13) shows th at q m = 2 mo d 4, con tradicting condition 2. The straight forwa rd enumeration has complexit y Ω(2 r ), but this can b e redu ced b y t wo d evices – 1. If (13) s h o ws that q m = 2 mo d 4 for some m < r / 2, we m a y use the fact that ǫ m in (12) dep end s only on q 0 , . . . , q 2 m to skip ov er a b lo c k of 2 r − 2 m − 1 n umbers N . By an argum ent similar to the h euristic argument f or the ord er of magnitude of ν ( r ), with su pp ort fr om empirical evidence for r ≤ 40, we conjecture th at this d evice reduces the complexit y of the en umeration to O r 2 2 r  3 4  r / 2 ! = O ( r 2 3 r / 2 ) . 2. Fix s , 0 ≤ s < r . Since ǫ r − m in (12) d ep ends only on q r − 2 m , . . . , q r , w e can tabulate those lo w-ord er bits b r − s · · · b r − 1 whic h do n ot n ecessarily lead to condition 2 b eing violated for some q r − m , 2 m ≤ s . In the enumeration we need only consider N with lo w-order bits in the table. W e conjecture that this r educes the complexit y of th e en umeration to O r 2 2 r  3 4  s/ 2 ! = O ( r 2 2 r − s 3 s/ 2 ) pro vided care is tak en to generate the table efficien tly . The tw o devices can b e combined, bu t they are n ot indep enden t. Th e complexit y of the com b ination is conjectured to b e O r 2 2 r  3 4  (6 r +5 s ) / 12 ! = O r 2 3 r / 2  3 4  5 s/ 12 ! , where the exp onent 5 s/ 12 (instead of s/ 2) refl ects the lac k of ind ep end en ce. In the computation of T able 2 w e us ed s ≤ 22 b ecause of memory constraints. The table size is O ( s 3 s/ 2 ) bits if th e table is stored as a list to tak e adv an tage of spars it y . Ac knowledgemen ts W e thank a referee for p oin tin g ou t an err or in the f orm ulation of Lemma 2 given in [2], and for pr o vid ing references to the classical literature. Th e ANU Su p ercomputer F acilit y p r o vided time on a F ujitsu VP 2200/10 for the d isco very of th e primitiv e trinomials mentio ned at th e end of Section 3. 10 References [1] S. L. Anderson, “Rand om num b er generators on v ector s u p ercompu ters and other adv anced arc h itectures”, SIAM R eview 32 (1990 ), 221-251. [2] R. P . 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Zierler, “On x n + x + 1 o v er GF (2)”, Information and Contr ol , 16 (1970) , 502-505. 12 r Q ( t ) 2 1 − t + t 2 5 1 − t − t 2 + t 4 + t 5 1 − t + t 2 + t 3 − t 4 − t 6 + t 9 9 1 − t + t 2 − t 3 − t 4 + t 8 + t 9 1 − t + t 2 − t 3 − t 4 − t 5 + t 6 + t 8 + t 9 10 1 − t + t 2 + t 3 + t 4 + t 6 − t 7 + t 9 + t 10 11 1 − t + t 2 − t 3 − t 4 + t 5 + t 6 − t 8 + t 11 12 1 − t + t 2 − t 3 − t 4 − t 8 + t 9 + t 11 + t 12 1 − t + t 2 − t 3 + t 4 − t 5 − t 6 + t 12 + t 13 1 − t + t 2 − t 3 + t 4 − t 5 − t 6 − t 7 + t 8 + t 12 + t 13 13 1 − t − t 2 − t 4 − t 6 + t 7 − t 8 + t 9 + t 10 + t 12 + t 13 1 − t + t 2 + t 3 + t 4 + t 5 + t 7 + t 9 − t 11 − t 12 + t 13 1 − t + t 2 + t 3 + t 4 + t 5 − t 8 − t 9 − t 11 − t 12 + t 13 1 − t + t 2 + t 3 − t 4 − t 6 − t 7 + t 8 + t 9 − t 11 + t 14 1 + t + t 3 − t 4 − t 5 + t 6 + t 7 + t 8 + t 9 − t 11 + t 14 14 1 − t − t 2 + t 3 − t 5 + t 6 + t 7 − t 8 − t 9 + t 13 + t 14 1 − t − t 2 − t 3 − t 5 + t 7 + t 9 + t 10 − t 11 + t 13 + t 14 1 − t − t 2 + t 4 − t 6 + t 8 + t 9 + t 10 + t 11 + t 13 + t 14 T able 1: Exceptional P olynomials of degree r ≤ 14 r ν ( r ) ¯ ν ( r ) r ν ( r ) ¯ ν ( r ) 1 0 0 2 1 79 0 .3923 2 1 1.78 22 94 0. 4390 3 0 0 2 3 231 0.4 837 4 0 0 2 4 129 0.4 650 5 1 0.70 25 428 0.4 388 6 0 0 2 6 448 0.4 615 7 0 0 2 7 883 0.4 964 8 0 0 2 8 635 0.4 218 9 3 0.83 29 1933 0.4 410 10 1 0. 30 30 14 70 0.4619 11 1 0. 13 31 43 80 0.4721 12 1 0. 22 32 31 25 0.4636 13 5 0. 33 33 72 32 0.4549 14 5 0. 37 34 88 62 0.4656 15 15 0.62 35 18870 0.47 92 16 12 0.58 36 10516 0.45 60 17 26 0.45 37 40082 0.45 47 18 18 0.41 38 39858 0.46 23 19 62 0.53 39 75370 0.47 12 20 34 0.45 40 54758 0.45 98 T able 2: Num b er of Exceptional P olynomials 13