New Constructions of Permutation Arrays

JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 1 Ne w Constructions of P ermutation Arrays Lizhen Y ang, K efei Chen, Luo Y uan Abstract A permutatio n array(permu tation code, P A) of leng th n and distan ce d , denoted by ( n, d ) P A, is a set of p ermutation s C from some fixed set of n elements such that th e Hamming distance between distinct mem bers x , y ∈ C is at least d . In this co rrespon dence, we present two constructio ns of P A from fractional polynom ials over finite field, a nd a construction of ( n, d ) P A fro m pe rmutation group with degree n an d min imal degree d . All these new con structions pro duces some n ew lower bounds for P A. Index T erms Code construction , permutatio n arra ys (P As), permuta tion cod e. I . I N T RO D U C T I O N Let Ω be an arbitrary non empty infinite set, and S y m (Ω ) denote the symmetri c group formed by the permutations over Ω . T wo dist inct permut ations x , y ∈ S y m (Ω ) hav e dist ance d if xy − 1 has exactly d unfixed points, in other words, there are exactly d p oints α ∈ Ω such that x ( α ) 6 = y ( α ) . This distance is also called Hamm ing distance. A permutation array(permutation code, P A) of length n and distance d , denoted b y ( n, d ) P A, is a set of permutations C from some fix ed set o f n elements such that the distance between distinct mem bers x , y ∈ C i s at least d . An ( n, d ) P A of size M is called an ( n, M , d ) P A. T he maximu m size of an ( n, d ) P A is denoted as P ( n, d ) . Manuscript receive d August 29, 2006. This work was supported by NS FC under grants 90104005 and 60573030 . Lizhen Y ang is with the department of computer science and engineering, Shanghai Jiaotong Uni versity , 800 DongChuan Road, Shanghai, 200420 , R.P . China (fax: 86-021-34 204221, email: lizhen yang@msn.com). Ke f ei Chen is with the department of computer science and engin eering, Shanghai Jiaotong Uni versity , 800 Dong Chuan Road, Shanghai, 200420 , R.P . C hina (fax: 86-021-3420 4221, email: Chen-kf@sjtu.edu.cn). Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 2 P As are somewhat studi es i n the 1970s. In 2000, an application by V inck [ ? ], [ ? ], [ ? ], [ ? ] of P As to a codin g/modulati on scheme for com munication over power lines has cre ated rene wed interest in P As . But the constructions and bounds for P As are fa r from completely . In thi s correspondence, we focus on the constructions of P As. Severa l papers h a ve been de voted to this problem. In 1974 Blak e [ ? ] presented a construction of P A based on the sharply k − tra n sitive groups. Usi ng thi s method , he constructed ( n, 3) , (11 , 8 ) , (1 2 , 8) , ( q , q − 1) , (1 + q , q − 1) P As with maximum sizes, where q is the power of a prime. In 2000 Kl ø ve [ ? ] gav e some constructi ons of ( n, n − 1) P A usin g t he l inear maps of a ring(commut ativ e with unity). In 2001 W adayama and V in ck [ ? ] presented som e m ultilevel con structions of P As with length n = 2 m . In 2002 Ding, et al. [ ? ] presented a method to con struct an r − bounded ( mn, mn − uv ) P A from an r − bounded ( n, n − u ) P A and an s -seperable ( m, m − v ) P A. In 2003, Chang, et al. [ ? ] presented the distance-preserving mapping to construct P As from binary codes. In 2004, Fu and Kløve [ ? ] presented two construction s of P As from P As and q − ary codes, Chu, et al. [ ? ] gav e several constructions, including construction from p ermutation polynomials ove r finite fields, Colbourn, et al. [ ? ] constructed ( n, n − 1) P As from mutu ally orthogonal latin squares of order n . In this correspon dence, we present two constructions of P A from fraction al polyn omials over finite fields, and a constructi on of ( n, d ) P A from permutation group with degree n and minimal degree d . Al l these new constructi ons yieds some new lower bounds for P A. In t he rest of this correspondence, we alw ays denote q as a power of prim e. I I . C O N S T RU C T I O N O F P A S F RO M F R AC T I O NA L P O L Y N O M I A L S Polynomials over finite fields are often used t o construct codes. In [ ? ], a class o f p olynomials called permutation polynom ials are directly applied to construction of P As. Let F q be a finite field of order q . A pol ynomial f over F q is said to be a permutation poly nomial (PP) i f t he induced map α 7→ f ( α ) from F q to its elf is bijective. Let N k ( q ) = { f : f ∈ F q [ x ] is a PP , ∂ ( f ) = k } . It was s hown in [ ? ] that the set of all of the permutation polynomials ov er F q with degree ≤ d forms an ( n, n − d ) P A. Theor em 1: [ ? ]. P ( q , q − d ) ≥ d X i =0 N i ( q ) . The s et of monic p ermutation polyn omials over F q with degree ≤ d + 1 forms an ( q , q − d ) P A also. Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 3 Theor em 2: [ ? ]. Suppose there are E monic permu tation polynomials over F q of d egree less than or equal t o d + 1 . Then P ( q , q − d ) ≥ E . In this section, we present two constructions of ( q , d ) and ( q + 1 , d ) based on the fractional polynomial s over finite fields. A. Constructi on of P As with length q Definition 1: A fractional polynom ial over F q is o f form f ( x ) g ( x ) , where f ( x ) , g ( x ) are poly- nomials over F q . T wo fractional polynom ials f 1 ( x ) g 1 ( x ) and f 2 ( x ) g 2 ( x ) are said to be equal, denoted as f 1 ( x ) g 1 ( x ) = f 2 ( x ) g 2 ( x ) , if and o nly if f 1 ( x ) = f 2 ( x ) , g 1 ( x ) = g 2 ( x ) . If g ( x ) is monic and ( f ( x ) , g ( x )) = 1 , then we say f ( x ) g ( x ) is sub-norm alized fractional polynomial. Let S F P ( q ) denote t he set of all sub-normalized fractional polyn omials over F q . For f ( x ) g ( x ) ∈ S F P ( q ) , define V  f ( x ) g ( x )  =      f ( α ) g ( α ) : α ∈ F q , g ( α ) 6 = 0      . Lemma 1: Suppose that φ = f 1 ( x ) g 1 ( x ) and ψ = f 2 ( x ) g 2 ( x ) are sub-normalized fractional permut ations over F q such that ∂ ( f 1 ( x ) g 2 ( x )) ≤ q − 2 , ∂ ( f 2 ( x ) g 1 ( x )) ≤ q − 2 . Then f 1 ( x ) g 2 ( x ) − f 2 ( x ) g 1 ( x ) = 0 . if and only if φ = ψ . Pr oof: The sufficiency is clear , we need only to prove the necessity . If f 1 ( x ) g 2 ( x ) − f 2 ( x ) g 1 ( x ) = 0 , then f 1 ( x ) g 2 ( x ) = f 2 ( x ) g 1 ( x ) , moreover ∂ ( f 1 ( x ) g 2 ( x )) ≤ q − 2 , ∂ ( f 2 ( x ) g 1 ( x )) ≤ q − 2 and ( f 1 ( x ) , g 1 ( x )) = 1 , then g 1 ( x ) | g 2 ( x ) . Similarly , g 2 ( x ) | g 1 ( x ) . Hence g 1 ( x ) = g 2 ( x ) because g 1 ( x ) and g 2 ( x ) are m onic. Then f 1 ( x ) = f 2 ( x ) follows immedi ately . QED. Definition 2: A P A-mapping for length q (for sho rt: an q − P AM) i s a m apping π : S F P ( q ) 7→ S y m ( F q ) f ( x ) g ( x ) 7→ ψ such that for each α ∈ F q , if A = { β ∈ F q : g ( β ) 6 = 0 , f ( β ) g ( β ) = α } 6 = Ø , then ψ − 1 ( α ) ∈ A , in ot her words, there exists β ′ ∈ A s atisfying ψ ( β ′ ) = α . Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 4 Pr opositi on 1: The number of q − P AMs i s at least Y φ ∈ S F P ( q ) ( q − V ( φ ))! . Pr oof: W e can con struct a q − P AM π as follows. For each φ = f ( x ) g ( x ) ∈ S P F ( q ) , according to the definition of q − P AM, we choose V ( φ ) m embers of { π ( φ )( α ) : α ∈ F q } determined by f ( α ) , g ( α ) , namely π ( φ )( α ) = f ( α ) g ( α ) , and set the other q − V ( φ ) members of { π ( φ )( α ) : α ∈ F q } to be any pos sibiliti es satisfyi ng { π ( φ )( α ) : α ∈ F q } = F q . There are at least ( q − V ( φ ))! possibili ties of { π ( φ )( α ) : α ∈ F q } . Thus we complete th e proof. QED. Lemma 2: min { s − s 1 , t − t 1 } + min { s − s 2 , t − t 2 } + max { s 1 + t 2 , s 2 + t 1 } ≤ s + t. Pr oof: For case max { s 1 + t 2 , s 2 + t 1 } = s 1 + t 2 , we have min { s − s 1 , t − t 1 } + min { s − s 2 , t − t 2 } + max { s 1 + t 2 , s 2 + t 1 } = min { s − s 1 , t − t 1 } + min { s − s 2 , t − t 2 } + ( s 1 + t 2 ) ≤ ( s − s 1 ) + ( t − t 2 ) + ( s 1 + t 2 ) = s + t. Similarly , for case that max { s 1 + t 2 , s 2 + t 1 } = s 2 + t 1 , t he statement ho lds als o. QED. Definition 3: Let s, t be no n-negati ve integer constants satisfying s + t ≤ q − 2 . Then we define S F P ( q , s, t ) be the set of all f ( x ) g ( x ) ∈ S F P ( q ) with ∂ ( f ( x )) = s ′ ≤ s , ∂ ( g ( x )) = t ′ ≤ t and q − V  f ( x ) g ( x )  ≤ min { s − s ′ , t − t ′ } . By definitio n, S F P ( q , s, 0) is equiv alent t o t he set of all permutation pol ynomials wit h degree ≤ s . In this poin t, S F P ( q , s, t ) can be regarded as a generalization of permutati on polynom ials, howe ver , S F P ( q , s, t ) are us ed to cons truct P As with help of q − P AM , rather than directly construction. Theor em 3: Let s, t be non-negativ e in teger constants satisfyin g s + t ≤ q − 2 . Then for any q − P AM π , { π ( φ ) : φ ∈ S F P ( q , s, t ) } is a ( q , | S F P ( q , s, t ) | , q − s − t ) P A. Pr oof: Let φ 1 = f 1 ( x ) g 1 ( x ) , φ 2 = f 2 ( x ) g 2 ( x ) ∈ S F P ( q , s, t ) with φ 1 6 = φ 2 , ∂ ( f 1 ( x )) = s 1 , ∂ ( g 1 ( x )) = t 1 , ∂ ( f 2 ( x )) = s 2 , ∂ ( g 2 ( x )) = t 2 . Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 5 Let r = |{ α ∈ F q : g 1 ( α ) 6 = 0 , g 2 ( α ) 6 = 0 , π ( φ 1 )( α ) = f 1 ( α ) g 1 ( α ) = π ( φ 2 )( α ) = f 2 ( α ) g 2 ( α ) }| , then r ≤      α ∈ F q : g 1 ( α ) 6 = 0 , g 2 ( α ) 6 = 0 , f 1 ( α ) g 1 ( α ) = f 2 ( α ) g 2 ( α )      = |{ α ∈ F q : g 1 ( α ) 6 = 0 , g 2 ( α ) 6 = 0 , f 1 ( α ) g 2 ( α ) − f 2 ( α ) g 1 ( α ) = 0 }| , by Lemma 1, f 1 ( x ) g 2 ( x ) − f 2 ( x ) g 1 ( x ) 6 = 0 , then r ≤ ∂ ( f 1 ( x ) g 2 ( x ) − f 2 ( x ) g 1 ( x )) ≤ max { s 1 + t 2 , s 2 + t 1 } . Then by the definitio n of q − P AM, t he number of ro ots of π ( φ 1 )( x ) − π ( φ 2 )( x ) = 0 in F q is at most ( q − V ( φ 1 ( x ))) + ( q − V ( φ 2 ( x ))) + r ≤ min { s − s 1 , t − t 1 } + min { s − s 2 , t − t 2 } + max { s 1 + t 2 , s 2 + t 1 } ≤ s + t, where the last i nequality follows from Lemm a 2. This yields t he theorem. QED. Cor ollary 1: For k ≤ q − 2 , P A ( q , q − k ) ≥ max {| S P F ( q , s, t ) | : s ≥ 0 , t ≥ 0 , s + t = k } ≥ k X i =0 N i ( q ) . Unfortunately , e ven the enumeration of permutation polynomials are f ar from complet e ( N k ( q ) has been known for q ≤ 5 [ ? ]), the enumeration of S F P ( q , s, t ) seem more difficulty than that of permutation p olynomials . But for small values of q , s, t , we can search by computer by checking all f ( x ) g ( x ) ∈ S F P ( q ) with ∂ ( f ( x )) ≤ s and ∂ ( g ( x )) ≤ t . Similar to case of permutation polynomial s, we can reduce the complexity of checking t asks by normalized their forms. Definition 4: A fractional polynomial f ( x ) g ( x ) over F q is said to be normalized if ( f ( x ) , g ( x )) = 1 , both f ( x ) and g ( x ) are monic, and when the degree s of f is not divisible by the characteristic of F q , the coeffic i ent of x s − 1 is 0. Let f ( x ) g ( x ) ∈ S F P ( q , s, t ) . For α, β ∈ F q , α 6 = 0 , then ψ = αf ( x + β ) g ( x + β ) ∈ S F P ( q , s, t ) is again a member of S F P ( q , s, t ) . By choosing α, β suitably , we can obtain ψ i n normalized form. For a give n normali zed fractional polynom ial f ( x ) g ( x ) , the num ber of distin ct such αf ( x + β ) g ( x + β ) is either q ( q − 1) or q − 1 , depending on whet her ( q , i ) = 1 for som e i ≥ 1 such t hat there i s a nonzero Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 6 coef ficient of x i . By this approach, we find som e new lower bou nds for P As by comput er . F or q ≤ 23 be prime and k ≤ 5 , the new lower bounds on P ( q , q − k ) are as fol lowing: P (19 , 16 ) ≥ 684 , P (19 , 1 5) ≥ 684 0 , P (19 , 14) ≥ 65 322 . Note: W e also find the two bounds P (19 , 16) ≥ 684 , P (1 9 , 1 5 ) ≥ 6840 [ ? ] from anoth er method. B. Constructi on of P As with length q + 1 Definition 5: A P A-mapping for length q + 1 (for short: an ( q + 1) − P A M) is a m apping π : S F P ( q ) 7→ S y m ( F q ∪ {∞} ) f ( x ) g ( x ) 7→ ψ such that: (1)for each α ∈ F q , A = { β : β ∈ F q , g ( β ) 6 = 0 , f ( β ) g ( β ) = α } 6 = Ø , ψ − 1 ( α ) ∈ A , in oth er words, t here exists β ′ ∈ A s atisfying ψ ( β ′ ) = α ; (2)if g ( x ) = 0 h as n o root in F q , t hen ψ ( ∞ ) = ∞ , else ψ − 1 ( ∞ ) is a root of g ( x ) = 0 i n F q . Pr opositi on 2: The number of ( q + 1) − P AMs is at least Y φ ∈ S F P ( q ) ( q − V ( φ ))! . Pr oof: W e can construct a ( q + 1) − P AM π as follows. For each φ = f ( x ) g ( x ) ∈ S P F ( q ) , according to the definiti on of ( q + 1 ) − P AM , we choose V ( φ ) members o f { π ( φ )( α ) : α ∈ F q } determined by f ( α ) , g ( α ) , nam ely π ( φ )( α ) = f ( α ) g ( α ) , and choose a members o f { π ( φ )( α ) : α ∈ F q ∪ {∞}} equal to ∞ , and then set the other q − V ( φ ) members of { π ( φ )( α ) : α ∈ F q ∪ {∞}} to be any possibili ties sati sfying { π ( φ )( α ) : α ∈ F q ∪ {∞}} = F q ∪ {∞} . There are at least ( q − V ( φ ))! possibili ties of { π ( φ )( α ) : α ∈ F q ∪ {∞}} . Th us we complete the proof. QED. Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 7 Definition 6: Let s, t, a, b be integer constants satis fying s ≥ 0 , t ≥ 0 , s + a ≥ 0 , t + b ≥ 0 , s + t ≤ q − 2 , s + t + a ≤ q − 2 , s + t + b ≤ q − 2 , s + t + a + b ≤ q − 2 . Then we define S F P ( q , s, t, a, b ) be set of all f ( x ) g ( x ) ∈ S F P ( q ) such t hat, suppo sing s ′ = ∂ ( f ( x )) , t ′ = ∂ ( g ( x )) , v = V  f ( x ) g ( x )  , for case that g ( x ) = 0 has roots in F q , s ′ ≤ s, t ′ ≤ t, and q − v ≤ min { s − s ′ , t − t ′ } + 1; for case that g ( x ) = 0 has n o root in F q , s ′ ≤ s + a, t ′ ≤ t + b and q − v ≤ min { s + a − s ′ , t + b − t ′ } . Theor em 4: Let s, t, a, b, d be integer const ants sati sfying s ≥ 0 , t ≥ 0 , s + a ≥ 0 , t + b ≥ 0 , s + t ≤ q − 2 , s + t + a ≤ q − 2 , s + t + b ≤ q − 2 , s + t + a + b ≤ q − 2 , d = min { q − s − t, q − s − t − a − b, q + 1 − s − t − max { a, b }} . Then for any ( q + 1) − P AM π , { π ( φ ) : φ ∈ S F P ( q , s, t, a, b ) } is a ( q + 1 , d ) P A with size | S F P ( q , s, t, a, b ) | . Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 8 Pr oof: Let φ 1 = f 1 ( x ) g 1 ( x ) , φ 2 = f 2 ( x ) g 2 ( x ) ∈ S F P ( q , s, t, a, b ) wit h φ 1 6 = φ 2 , s 1 = ∂ ( f 1 ( x )) , t 1 = ∂ ( g 1 ( x )) , v 1 = V ( φ 1 ) , s 2 = ∂ ( f 2 ( x )) , t 2 = ∂ ( g 2 ( x )) , v 2 = V ( φ 2 ) . Let r = |{ α ∈ F q : g 1 ( α ) 6 = 0 , g 2 ( α ) 6 = 0 , π ( φ 1 )( α ) = f 1 ( α ) g 1 ( α ) = π ( φ 2 )( α ) = f 2 ( α ) g 2 ( α ) }| , then r ≤      α ∈ F q : g 1 ( α ) 6 = 0 , g 2 ( α ) 6 = 0 , f 1 ( α ) g 1 ( α ) = f 2 ( α ) g 2 ( α )      = |{ α ∈ F q : g 1 ( α ) 6 = 0 , g 2 ( α ) 6 = 0 , f 1 ( α ) g 2 ( α ) − f 2 ( α ) g 1 ( α ) = 0 }| By Lemma 1, f 1 ( x ) g 2 ( x ) − f 2 ( x ) g 1 ( x ) 6 = 0 , then r ≤ ∂ ( f 1 ( x ) g 2 ( x ) − f 2 ( x ) g 1 ( x )) − z ≤ max { s 1 + t 2 , s 2 + t 1 } − z , where z = { α ∈ F q : g 1 ( α ) = g 2 ( α ) = 0 } . Now we are ready to find the upper bound on the number of roots of π ( φ 1 )( x ) − π ( φ 2 )( x ) = 0 in F q ∪ {∞} , which is deno ted as R . W e discus s in four cases: Case I): Both g 1 ( x ) = 0 and g 2 ( x ) = 0 have root s in F q . W e further discuss in two s ubcases: Subcase 1): g 1 ( x ) = 0 and g 2 ( x ) = 0 ha ve at least a common root in F q , namely z ≥ 1 . By the definition of ( q + 1) − P AM , we have R ≤ z + ( q − v 1 − z ) + ( q − v 2 − z ) + r + 1 ≤ z + ( q − v 1 − z ) + ( q − v 2 − z ) + max { s 1 + t 2 , s 2 + t 1 } − z + 1 = ( q − v 1 ) + ( q − v 2 ) + max { s 1 + t 2 , s 2 + t 1 } − 2 z + 1 ≤ min { s − s 1 , t − t 1 } + 1 + min { s − s 2 , t − t 2 } + 1 + max { s 1 + t 2 , s 2 + t 1 } − 1 ≤ s + t + 1 . where the last i nequality follows from Lemm a 2. Subcase 2): z = 0 . T hen i f α ∈ Z q is a root of g 1 ( x ) = 0 satisfyi ng π ( φ 1 )( α ) = ∞ th en π ( φ 2 )( α ) 6 = ∞ , whereas if α ′ ∈ Z q is a root of g 2 ( x ) = 0 satisfying π ( φ 2 )( α ′ ) = ∞ then π ( φ 1 )( α ′ ) 6 = ∞ . So we ha ve R ≤ ( q − v 1 − 1) + ( q − v 2 − 1) + r + 1 ≤ min { s − s 1 , t − t 1 } + min { s − s 2 , t − t 2 } + max { s 1 + t 2 , s 2 + t 1 } + 1 ≤ s + t + 1 . Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 9 Case II): Both g 1 ( x ) = 0 and g 2 ( x ) = 0 hav e no root, then z = 0 . By the definition o f ( q + 1) − P AM, we ha ve R ≤ ( q − v 1 ) + ( q − v 2 ) + r + 1 ≤ min { s + a − s 1 , t + b − t 1 } + min { s + a − s 2 , t + b − t 2 } + max { s 1 + t 2 , s 2 + t 1 } + 1 ≤ s + a + t + b + 1 . Case III): g 1 ( x ) = 0 has roots in F q while g 2 ( x ) = 0 has no root. Then z = 0 . If α ∈ Z q is a root of g 1 ( x ) = 0 satisfying π ( φ 1 )( α ) = ∞ then π ( φ 2 )( α ) 6 = ∞ , and π ( φ 1 )( ∞ ) 6 = ∞ whi le π ( φ 2 )( ∞ ) = ∞ . Then b y the definition of ( q + 1) − P AM , we have R ≤ ( q − v 1 − 1) + ( q − v 2 ) + r ≤ min { s − s 1 , t − t 1 } + min { s + a − s 2 , t + b − t 2 } + max { s 1 + t 2 , s 2 + t 1 } ≤ min { s − s 1 , t − t 1 } + min { s − ( s 2 − a ) , t − ( t 2 − b ) } + max { s 1 + ( t 2 − b ) , ( s 2 − a ) + t 1 } + max { a, b } ≤ s + t + ma x { a, b } . Case IV): g 1 ( x ) = 0 has no roots while g 2 ( x ) = 0 has roots in F q . It can b e proved that R ≤ s + t + max { a, b } similar to Case III. Now we can conclude that { π ( φ ) : φ ∈ S F P ( q , s, t, a, b ) } is a ( q + 1 , d ) P A with size | S F P ( q , s, t, a, b ) | , wh ere d = q + 1 − R ≥ min { q − s − t, q − s − t − a − b, q + 1 − s − t − max { a, b }} . QED. Comparing the definitions of S F P ( q , s, t ) with S F P ( q , s, t, a, b ) , we find S F P ( q , s, t ) ⊆ S F P ( q , s, t, 0 , 0) . This in conju nction with Theorem 4 implies t he following Corollary . Cor ollary 2: For k + 1 ≤ q − 2 , P A ( q + 1 , q − k ) ≥ max {| S F P ( q , s, t, a, b ) | : s + t = k , s ≥ 0 , t ≥ 0 , s + a ≥ 0 , t + b ≥ 0 , ( a, b ) ∈ { (0 , 0) , (1 , − 1 ) , ( − 1 , 1) }} ≥ max {| S P F ( q , s, t ) : s ≥ 0 , t ≥ 0 , s + t = k } . As the case of S F P ( q , s, t ) , th e enumeration of S F P ( q , s, t, a, b ) is difficulty to determi ned, while for s mall values of q , s, t , we can find by computer by checkin g all f ( x ) g ( x ) ∈ S F P ( q ) Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 10 with ∂ ( f ( x )) ≤ s + max { 0 , a } , ∂ ( f ( x )) ≤ t + max { 0 , b } . The complexity of checking task can be also reduced by onl y checking the normalized forms. Let f ( x ) g ( x ) ∈ S F P ( q , s, t, a, b ) . For α, β ∈ F q , α 6 = 0 , then ψ = αf ( x + β ) g ( x + β ) ∈ S F P ( q , s, t, a, b ) is again a memb er of S F P ( q , s, t, a, b ) . By choosing α , β suitably , we can obtain ψ in n ormalized form. For a giv en normalized fractional polynomial f ( x ) g ( x ) , t he number of di stinct such αf ( x + β ) g ( x + β ) is either q ( q − 1) or q − 1 , depending on whether ( q , i ) = 1 for som e i ≥ 1 such that there is a n onzero coefficient of x i . By t his app roach, we find some new lower bounds for P A by com puter . For q ≤ 23 be pri me and k ≤ 5 , the n e w lower bounds on P ( q + 1 , q − k ) are as following: P (18 , 14 ) ≥ 9520 , P (20 , 1 4) ≥ 123 8 04 , P (24 , 20) ≥ 2378 2 . I I I . C O N S T RU C T I O N O F ( n, d ) P A S F RO M P E R M U TA T I O N G R O U P S W I T H D E G R E E n A N D M I N I M A L D E G R E E d Let G be a finite permutation group with a action on a set Ω . The order of G is defined as the cardin ality of G and the de gree of G is defined as the cardinality of Ω . If g ∈ G , then the degree of g on Ω is the numb er of points moved by g . Th e minimal degree of G i s the minimum degree of a nontrivial element in G . G has fixi ty f if nontrivial elements of G fixes ≤ f po ints, and there is a n ontrivial element of G fixing exactly f points. Thus the minimal d egree of a permutation group of degree n and fixity f is n − f . By definition s, for g 1 , g 2 ∈ G , the dis tance between g 1 and g 2 is the degree of g 1 g − 1 2 , this yi elds the following theorem immediately . Theor em 5: Let G be a permutation grou p with degree n and m inimal degree ≥ d . Then G form an ( n, d ) P A. Then from the knowledge of permutation groups , we can obtain a lot of P As for give n lengths and distances. Example 1: Frobenius groups hav e fixi ty one, t hen Frobenius group of degree n forms an ( n, n − 1) P A [ ? , p.85 ]. Zassenhaus group s have fixity two, then Zassenhaus group wit h degree n forms an ( n, n − 2) P As. The minim al de g ree of a proper primitive permutation groups of degree n is at l east 2( √ n − 1) [ ? ], then a proper primi tiv e permu tation group of degree n form s an ( n, 2( √ n − 1)) P A. Example 2: Let m ≥ 5 , n = m ( m − 1) / 2 , and d = 2 m − 4 , and perm utation group G be the action of S m on the set of 2 − sets o f { 1 , 2 , . . . , m } . Thi s action i s primitive of degree Nov ember 21, 2018 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, A UGUST 2006 11 n = m ( m − 1) / 2 with minimal degree d = 2 m − 4 [ ? , Exercise 3.3.5, p.77 ]. Then G is an ( n, d ) P A with | G | ≥ exp( √ 2 n log √ 2 n − √ 2 n ) [ ? , Exercise 5.3.4, p.155 ]. Example 3: Let G be th e affine group AGL d ( q ) acts as a permutation on the affine space of dimension d over a field of q elements. Then G is an ( q d , q d − q d − 1 ) P A [ ? , Example 5.4 .1, p.158] of size q d ( d +1) / 2 ( q d − 1)( q d − 1 − 1) . . . ( q − 1) . Particularly , for q = 2 , G i s a (2 d , 2 d − 1 ) P A wit h size 2 d ( d +1) / 2 (2 d − 1)(2 d − 1 − 1) . . . ( 2 − 1) wh ich is 2 ( d +1) / 2 times the size 2 d (2 d − 1)(2 d − 1 − 1) . . . ( 2 − 1) of P A con structed in [ ? ]. Some n e w lowe r bou nds for P As are obt ained below . Lemma 3: P (24 , 16) ≥ 24482 3 040 , P (23 , 16) ≥ 10 200960 , P (22 , 1 6) ≥ 44 3 520 . Pr oof: The fixit y of Mathieu group M 24 is at most 8 [ ? , p.310], then M 24 is a (24 , 16) P A of size | M 24 | = 2 10 · 3 3 · 5 · 7 · 11 · 23 = 24482 3040 [ ? , T able 6.1., P .204]. Since Mathieu group M 23 is a on e-point st abilizer of M 24 , then M 23 is a (23 , 16) P A with size | M 23 | = 2 7 · 3 2 · 5 · 7 · 11 · 23 = 10200960 [ ? , T able 6.1., P .204]. Since Mathi eu group M 22 is a two-point st abilizer of M 24 , then M 22 is a ( 22 , 16) P A with size | M 22 | = 2 7 · 3 2 · 5 · 7 · 11 = 443520 [ ? , T able 6.1 ., P .204]. QED. The permutati on group G actin g on a set Ω wi th | Ω | = n is said to be sharply k -t ransitive if, for any two ordered k subsets of Ω , say { i 1 , i 2 , . . . , i k } and { j 1 , j 2 , . . . , j k } there exists exactly one element σ ∈ G such that σ ( i l ) = j l , l = 1 , . . . , k . The sharply k -t ransitive g roup of degree n has been proved in [ ? ] th at i t is an ( n, n − k + 1) P A of size n ! / ( n − k )! . Indeed, a permutation group G with | G | = n ! / ( n − k )! of d egree n i s an ( n, n − k + 1) P A if and o nly if it i s sharply k -transitive. Theor em 6: A permut ation grou p G with order n ! / ( n − k )! and degree n is an ( n, n − k + 1) P A if and only if it is sh arply k -transitive. Pr oof: W e need only to prove the necessary . Suppose that G acts on Ω . F o r any two ordered k subsets of Ω , say { i 1 , i 2 , . . . , i k } , { j 1 , j 2 , . . . , j k } , there exists at mo st one σ ∈ G such that σ ( i l ) = j l , l = 1 , . . . , k , (1) since t wo disti nct s uch perm utations would hav e dist ance ≤ n − k . There are n ! / ( n − k )! o rdered k subs ets of Ω , thi s means t here exists at least one element σ ∈ G satisfying cond ition (1). Hence G is sharply k -transitiv e. QED. mds November 18, 2002 Nov ember 21, 2018 DRAFT