New Lower Bounds on Sizes of Permutation Arrays

JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 1 Ne w Lo wer Bounds on S izes of Permutation Arrays Lizhen Y ang , Kefei Chen, Luo Y uan Abstract A permutatio n arr ay(or code) of length n and distance d , denoted by ( n, d ) P A, is a set of permu - tations C from some fixed set of n elements such that the H amming distance be tween distinct memb ers x , y ∈ C is at least d . Let P ( n, d ) d enote the maximum size of an ( n, d ) P A. This co rrespon dence focuses on th e lower b ound on P ( n, d ) . Fi r st we g iv e three im provements over the Gilbert-V arsha mov lower bounds on P ( n, d ) b y applying the graph theorem framework presen ted by Jiang and V ardy . Next we sho w anoth er two new improved boun ds by consider ing the covered balls intersections. Fin ally som e new lower bounds for certain values of n and d are given. Index T erms permutatio n arrays (P As), p ermutation codes, lower boun ds. I . I N T RO D U C T I O N Let Ω be an arbit rary nonempty infinite set. T wo disti nct permutations x , y ov er Ω have distance d if xy − 1 has exactly d unfixed points. A permutation array(permutatio n cod e, P A) of length n and distance d , denoted by ( n, d ) P A, is a set of permut ations C from s ome fixed s et of n elements such that the distance between dis tinct m embers x , y ∈ C is at least d . An ( n, d ) P A of si ze M is called an ( n, M , d ) P A. T he m aximum size of an ( n, d ) P A is denoted as P ( n, d ) . Manuscript receive d June 1, 2006. This work w as supported by NFSE u nder grants 90 104005 and 605 73030. Lizhen Y ang is with the de partment of computer sc i ence and engineering, Shanghai Jiao tong University , 800 DongChuan Road, Shang hai, R.P . China (fax: +86-21-342042 21, email:lizhen yang@msn.com). Ke f ei Chen is with the department of computer science and engin eering, Shanghai Jiaotong Uni versity , 800 DongChuan Road, Shanghai, R.P . China (f ax: +86-21-3420 4221, email: Chen-kf@sjtu.edu.cn). September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 2 P As are somewhat studies in the 1970s. A recent application by V inck [ ? ], [ ? ], [ ? ], [ ? ] of P As t o a coding/ modulation scheme for communication over power lin es has created rene wed interest in P As. But there are s till m any problem s u nsolved i n P As, e.g. one of the essenti al problem is to compu te t he values of P ( n, d ) . It’ s known that determini ng the exactly values of P ( n, d ) is a difficult task, except for special cases, it can be only t o establish some lower bounds and upper b ounds on P ( n, d ) . W e shall stu dy how to determine lower bound on P ( n, d ) i n this correspondence, and give some new bounds. A. Concepts and Notati ons In t his sub section, we introduce concepts and notati ons that will be us ed t hroughout the correspondence. Since for t wo sets Ω , Ω ′ of the same size, the s ymmetric g roups S y m (Ω) and S y m (Ω ′ ) formed by the permutations ov er Ω and Ω ′ respectiv ely , under compositi ons of mapping s, are isomorphic, we need only to con sider the P As over Z n = { 0 , 1 , . . . , n − 1 } and write S n to denote th e special group S y m ( Z n ) . In the rest o f the correspondence, without special pointed out, we alw ays assume that P As are over Z n . W e also write a permutati on a ∈ S n as an n − tuple ( a 0 , a 1 , . . . , a n − 1 ) , where a i is the i mage of i under a for each i . Especially , we write t he identi cal permutation (0 , 1 , . . . , n − 1) as 1 for con venience. The Hamm ing distance d ( a , b ) between two n − tuples a and b is the number of positio ns where they differ . Then the distance between any two permut ations x , y ∈ S n is equiv alent t o their Hamming distance. Let C be an ( n, d ) P A. A permutatio n in C i s also call ed a codeword of C . F or con venience for discussion, without loss of generality , we always assume that 1 ∈ C , and the indies of an n − tuple (vector , array) are started by 0 . The support of a binary vector a = ( a 0 , a 1 , . . . , a n − 1 ) ∈ { 0 , 1 } n is defined as th e set { i : a i = 1 , i ∈ Z n } , and the weight of a i s the size of it s support, namely the nu mber of ones in a . The support of a permutation x = ( x 0 , x 1 , . . . , x n − 1 ) ∈ S n is d efined as the set o f the poi nts not fixed by x , namely { i ∈ Z n : x i 6 = i } = { i ∈ Z n : x ( i ) 6 = i } , and the weigh t of x , denoted as w t ( x ) , is d efined as the size of i ts suppo rt, namely the num ber of points in Z n not fixed by x . For an ( n, d ) P A C , we say th at a permu tation a ∈ S n is covered by a code word x ∈ C , if d ( a , x ) < d . The set of p ermutations i n S n cove red by x ∈ C is denoted as B ( x ) and called the covere d ball of x . A derangement of order k is an element of S k with no fixed point s. September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 3 Let D k be the number of derangements of order k , with th e con vention t hat D 0 = 1 . Th en D k = k ! P k i =0 ( − 1) k k ! =  k ! e  , where [ x ] is the nearest integer functi on, and e is the base of the natural logarithm . Then | B ( x ) | = V ( n, d − 1) = d − 1 X i =0  n i  D i . (1) For an arbitrary perm utation x ∈ S n , d ( x , C ) stands for the Hammi ng distance b etween x and C , i. e., d ( x , C ) = min c ∈ C d ( x , c ) . A permutati on x is called covered by C if d ( x , C ) < d . The set of permutations co vered by C is denoted as B ( C ) and called the co vered ball of C . C l early , B ( C ) = ∪ c ∈ C B ( c ) . Finally , we d efine P [ n, d − 1] as the m aximum si ze of the subset Γ of S n such that the dist ance between two d istinct permutations in Γ i s d − 1 at most. W e wi ll show t hat P ( n, d ) hav e close relations with P [ n, d − 1] . B. Pr evious W ork on th e Lower Bounds on P ( n, d ) By the definitions of P ( n, d ) , it is easy to ob tain t he following well-kn own elementary consequences that are firstly appeared in [ ? ] and summ arized in [ ? ]. Pr opositi on 1: P ( n, 2) = n ! , (2) P ( n, 3) = n ! / 2 , (3) P ( n, n ) = n, (4) P ( n, d ) ≥ P ( n − 1 , d ) , P ( n, d + 1) , (5) P ( n, d ) ≤ nP ( n − 1 , d ) , (6) P ( n, d ) ≤ n ! / ( d − 1)! . (7) A latin square of order n i s an ( n, n ) P A. T wo latin s quares L = ( L i,j ) and L ′ = ( L ′ i,j ) are orthogonal if { ( L i,j , L ′ i,j ) : 1 ≤ i, j ≤ n } = { 1 , 2 , . . . , n } 2 . The following proposi tion was proved by Colbourn et al. [ ? ]. Pr opositi on 2: [ ? ]. If there are m mutually orthogonal latin squares of order n , then P ( n, n − 1) ≥ mn . In particular , if q is a prime-power , then P ( q , q − 1) = q ( q − 1) . It was po inted out by Frankl and Deza [ ? ] that the existence of a sharply k − transitive group acting on a set of size n is equiva l ent to a maximum ( n, n − k + 1) P A. It is well known that t he September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 4 Normalized Permutation Polynomials q restriction T otal x any q q ( q − 1) x 2 q ≡ 0 mod 2 q ( q − 1) x 3 q 6≡ 1 mod 3 q 2 ( q − 1) or q ( q − 1) x 3 − ax ( a not a square) q ≡ 0 mod 3 q ( q − 1) 2 / 2 x 4 ± 3 x q = 7 2 q 2 ( q − 1) x 4 + a 1 x 2 + a 2 x (if only root in F q is 0) q ≡ 0 mod 2 1 3 q ( q − 1)( q 2 + 2) x 5 q 6≡ 1 mod 5 q 2 ( q − 1) or q ( q − 1) x 5 − ax ( a not a fourth po wer) q ≡ 0 mod 5 3 4 q ( q − 1) 2 x 5 + ax ( a 2 = 2) q = 9 2 q 2 ( q − 1) x 5 ± 2 x 2 q = 7 2 q 2 ( q − 1) x 5 + ax 3 ± x 2 + 3 a 2 x ( a not a square) q = 7 q 2 ( q − 1) 2 x 5 + ax 3 + 5 − 1 a 2 x ( a arbitrary) q ≡ ± 2 mo d 5 q 3 ( q − 1) x 5 + ax 3 + 3 a 2 x ( a not a square) q = 13 1 2 q 2 ( q − 1) 2 x 5 − 2 ax 3 + a 2 x ( a not a square) q ≡ 0 mod 5 1 2 q 2 ( q − 1) 2 T AB LE I N O R M A L I Z E D P E R M U T AT I ON P O LYN O M I A L S W I T H D E G R E E d ≤ 5 group P GL (2 , q ) , consistin g of fractional li near transformation s x 7→ ( ax + b ) / ( cx + d ) , ad − bc 6 = 0 , is s harply 3 − transitive acting on X = F q ∪ {∞} , and the Math ieu groups M 11 and M 12 are sharply 4 − and 5 − transitive on sets of size 11 and 12, respectiv ely . Pr opositi on 3: [ ? ]. If q is a prime-power , then P ( q + 1 , q − 1) = ( q + 1) q ( q − 1) . Addition ally , P (11 , 8) = 11 · 10 · 9 · 8 and P (12 , 8) = 12 · 11 · 10 · 9 · 8 . Let F q be a finite field of order q . A polynomial f over F q is a p ermutation polynomial if the mapping it defines is one-to-on e. Let N d ( q ) denote the number of t he permutation po lynomials over F q of given degree d ≥ 1 . By a direct constructio n of P As from permutation polynomi als, Chu et al. [ ? ] proved the fol lowing connection between P ( q , q − d ) and N i ( q ) . Pr opositi on 4: [ ? ]. L et q be a prime power . Then P ( q , q − d ) ≥ P d i =1 N i ( q ) . Unfortunately , not much is kn own about permutati on polynom ials. Whil e their classification and enumeratio n are far from complete, ev erythi ng is known for d < 6 . The normalized permutation polynom ials with degree d ≤ 5 , together with t he total produced by each class are given in T able I, sum marized by Chu et al. [ ? ] according to the table in [ ? ]. September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 5 By a si mply observation, Chu et al. [ ? ] also proved another connection between permutation polynomial s and P ( q , q − d ) . Pr opositi on 5: [ ? ]. Suppose q is a prime-power and that there are M moni c permutation polynomial over F q of degree less t han o r equal to d + 1 . Then P ( q , q − d ) ≥ M . The fol lowing result is imm ediately go tten from Proposition 5 and T abl e I. Cor ollary 1 : [ ? ]. If q is a prim e-power , q 6≡ 1 mod 3 1 , then P ( q , q − 2) ≥ q 2 . In [ ? ], T .Kløve prov ed the following lower bound on P ( n, n − 1) by generalized t he approach in [ ? ]. Pr opositi on 6: Let n = P u i =1 p c i i be the standard factorization of n , and let θ ( n ) = min { p c i i | 1 ≤ i ≤ u } . (8) Then for all n > 1 we have P ( n, n − 1) ≥ n ( θ ( n ) − 1) . The oth er explicit construction s leading to lower bo unds on P ( n, d ) are listed bel ow . In [ ? ], C. Ding, et al. presented a c o nstruction of ( mn, mn − 1) P A with s ize m | C | from an r − bounded ( n, n − 1) P A and an s − separable ( m, m − 1 ) P A. In [ ? ], Fu and Kløve presented two constructions of P As with length q n from ( n, d ; q ) codes and ( n, d ) P As. In [ ? ], Chu et al. proposed a recursi ve construction o f P A and used this constructi on to derive a lower boun d on P ( n, 4) and a lower bound that P ( n, n − 2) ≥ 2 q ( q − 1) , wheneve r n = q + q ′ is a sum of two prim e powers with 0 ≤ q ′ − q ≤ 2 . For certain s mall values of n and d , t he lower bounds on P ( n, d ) can be also directly deter- mined by compu tational constructions . Deza and V anston e [ ? ] first used computer constructi on to prove P (6 , 5) = 18 and P (1 0 , 9) ≥ 32 . In [ ? ], Chu et al. presented three comp utational methods of clique s earch, greedy algorit hm and automorph isms, and got some ne w lower bound s for certain values of n and d . For n ≤ 13 and certain values of n ≥ 14 and d , the best pre vious lower bounds on P ( n, d ) are summ arized in [ ? ]. The only g eneral lo wer bound o n P ( n, d ) is the Gi lbert-V arshmov boun d, whi ch is derived in a simi lar w ay as the Gilbert-V arshm ov bound for binary codes. Let A ( n, d ) be the maxi mum 1 In [ ? ], q 6≡ 1 mo d 3 is replaced by q 6≡ 2 mo d 3 , but b y T able I it should be q 6≡ 1 mo d 3 . September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 6 size of an ( n, d ) b inary cod e, then A ( n, d ) ≥ 2 n V 2 ( n, d − 1) , where V 2 ( n, d − 1) i s the volume of a sphere in { 0 , 1 } n of radius d − 1 , that i s, V 2 ( n, d − 1) = d − 1 X i =0  n i  . (9) Similarly , the Gilbert-V arshamov bound [ ? ] on P ( n, d ) is as follows: P ( n, d ) ≥ n ! V ( n, d − 1) . C. Our New Resu lts In this correspondence, we first give three improvements over the Gilbert-V arsham ov lower bounds on P ( n, d ) by using th e graph theorem frame work presented by Jiang and V ardy in [ ? ]. In 2004, J iang and V ardy presented a graph theorem framew ork which may lead to improvements over Gi lbert-V arshamov bound for codes if the corresponding Gilbert-V ash amov graphs are sparse. They were successful to asymptoti cally improve the Gilbert-va rsh amov bound on size of binary codes by a f actor of n when d is proportional to n , namely , d = α n for some pos itive constant α . Recently , V u and W u [ ? ] generalized the resu lts of Jiang and V ardy to q -ary codes. Employing the graph t heorem framework, we also establish the following three ne w theorems in lower bounds on P ( n, d ) . Theor em 1: For x ∈ R , let ⌈ x ⌉ + denote the small est nonnegative in teger m wit h m ≥ x . Giv en positive integers n and d , wi th d ≤ n , let E ( n, d ) denote the following quanti ty: E ( n, d − 1) = 1 6 d − 1 X i =2 d − 1 X j =2  n i  D i L i,j where L i,j = min( i,j ) X k = ⌈ i + j − d +1 2 ⌉ + min { d +2 k − i − j − 1 ,k } X l =0  i k  n − i j − k  k l  ( l + j − k )! . Then P ( n, d ) ≥ n ! 10 V ( n, d − 1 ) (log 2 V ( n, d − 1) − 1 / 2 lo g 2 E ( n, d − 1)) (10) Theor em 2: Let α be a constant s atisfying 0 < α < 1 / 2 . Then t here is a posit iv e constant c depending on α su ch t hat the following holds. For d = αn , P ( n, d ) ≥ c n ! V ( n, d − 1) log 2 V 2 ( n, d − 1) . September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 7 Theor em 3: Let α be a constant satisfyin g 0 < α < 1 . Then there is a positive constant c depending on α su ch t hat the following holds. For d = n α , P ( n, d ) ≥ c n ! V ( n, d − 1) log 2 V ( n, d − 1) . Secondly , another t wo improvements over Gilbert-V arsh amov lo w er bound s are est ablished by considering the covered balls intersection s. W e will prove in section III that P ( n, d ) ≥ 2 · n ! V ( n, d − 1) + P [ n, d − 1] . Let C ′ be an ( n, M , d ) P A, then w e will prove in section III t hat P ( n, d ) ≥ n ! M | B ( C ′ ) | . Our third con tribution is to gi ve so me ne w lower bounds on P ( n, d ) for certain cases of n and d based on t he t wo new relati ons: for n ≥ d > 3 P ( n − 1 , n − 3) ≥ P ( n, d ) , (11) and for n ≥ d > 2 P ( n − 1 , d − 2) ≥ 2 n P ( n, d ) . I I . I M PR OV E D G I L B E R T - V A R S H A M OV B O U N D B Y G R A P H T H E O R E T I C F R A M E W O R K W e first recall a few basic notions from graph t heory . A graph G consist s of a (finite) set V ( G ) of vertices and a set E ( G ) of edges, where an edge is a (non-ordered) pair ( a, b ) with a, b ∈ V ( G ) . If a and b f orm an edge, we say that th ey are adjacent. The set of all n eighbors of a verte x v i s denoted as N ( v ) and called the neighb orhood of v . The degree of a v ertex v ∈ V ( G ) , denoted as de g ( v ) , is defined as deg( v ) = | N ( v ) | . The graph is D -regular if t he de g ree of ever y verte x equals D . A subset I of V ( G ) is an ind ependent set if it does not contain any edge. The independence number of G i s the size of the largest independent set in G , and is denoted as α ( G ) . Definition 1: Let n and d ≤ n be positive i ntegers. The corresponding Gi lbert graph G 2 over { 0 , 1 } n is defined as following: V ( G 2 ) = { 0 , 1 } n and { u , v } ∈ E ( G 2 ) i f and onl y if 1 ≤ d ( u , v ) ≤ d − 1 . Definition 2: Let n and d ≤ n be positive integers. The corresponding Gilbert graph G P over S n is defined a s follo w ing: V ( G P ) = S n and { u , v } ∈ E ( G P ) if a n d only i f 1 ≤ d ( u , v ) ≤ d − 1 . September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 8 Then clearly , an ( n, d ) binary code is an independent set in the Gilbert graph G 2 . Con versely , any independent set in G 2 is an ( n, d ) binary code. This means A ( n, d ) = α ( G 2 ) . Similarly , P ( n, d ) = α ( G P ) . By appl ying a simpl e observation on graph to a graph theorem in Bollob ´ as [ ? , Lemma 15, p.296], Jiang and V ardy [ ? ] prove the foll owing theorem. Theor em 4: [ ? ]. Let G be a graph wit h m aximum degree at most D , and suppos e that for all v ∈ V ( G ) , the subgraph of G ind uced by the neighborho od of v has at most T edges. Then α ( G ) ≥ n ( G ) 10 D (log 2 D − 1 / 2 log 2 ( T / 3)) , where n ( G ) is the num ber of vertices of G . W e consid er t he Hammin g sphere graph G S P over S n that is t he subgraph of the Gi lbert graph G P over S n induced by the neighborho od N ( 1 ) of the vertex 1 ∈ V ( G P ) . Clearly , t he subgraph induced in t he Gil bert graph over S n by t he neigh borhood of any other vertex in G P is isomo rphic to G S P . T o deriv e an upper bound for the edges of G S P , we need to consider the H amming sphere graph G S 2 over { 0 , 1 } n , that is the s ubgraph of the Gilbert graph G 2 over { 0 , 1 } n induced by the n eighborhood N ( 0 ) of the vertex 0 ∈ V ( G 2 ) . For the sake of clearer presentation, we define T = | E ( G S P ) | , D = | V ( G S P ) | = V ( n, d − 1) − 1 , T ′ = | E ( G S 2 ) | , D ′ = | V ( G S 2 ) | = V 2 ( n, d − 1) − 1 , where V ( n, d − 1) and V 2 ( n, d − 1) are defined by (1) and (9) respectiv ely . Lemma 1: For any x ∈ S n of weight i , there are at most L i,j = min( i,j ) X k = ⌈ i + j − d +1 2 ⌉ + min { d +2 k − i − j − 1 ,k } X l =0  i k  n − i j − k  k l  ( l + j − k )! permutations of weight j wi th dist ance less than d to x , where ⌈ x ⌉ + denotes the smallest nonnegativ e integer not less than x . Pr oof: W ithout loss of generality , sup pose the s upport of x is X = { 0 , 1 , . . . , i − 1 } . Let y be an arbitrary permutation with weight of j and support of Y , having dis tance less t han d to x . Let Z = X ∩ Y and R = { r ∈ X ∩ Y : x ( r ) 6 = y ( r ) } . Then it follows from d − 1 ≥ d ( x , y ) = | X | + | Y | − 2 | X ∩ Y | + | R | = i + j − 2 | Z | + | R | that | R | ≤ d + 2 | Z | − i − j − 1 . (12) Since R ≥ 0 , | Z | ≥ i + j − d +1 2 by (12). There are at most  i k  n − i j − k  candidates of Y such that | Z | = k , and for each candidate of Y satisfying | Z | = k there are at most  k l  ( l + j − k )! corresponding permutations sati sfying | R | = l . Therefore the lem ma follo ws immediately . QED. September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 9 Lemma 2: T ≤ 1 2 d − 1 X i =2 d − 1 X j =2  n i  D i L i,j (13) Pr oof: Since G S P has  n i  D i vertices of weight i , and t here has no vertices with weight 1 , (13) follows immediately from Lemma 1. QED. Comparing the fore g oing expression for the upper bound on T with the expression for E ( n, d − 1) in Theorem 1 , we see that E ( n, d − 1) ≥ T 3 . Th us Lemma 2 in conjunction with Theorem 4 induces (10). This complet es th e proo f of Theorem 1. Now we t urn to the asymptotic boun ds on T whi ch will in turn in duce t he asy mptotic bounds on P ( n, d ) . Instead of u sing the upper bound on T presented in Lemm a 2, we use th e foll owing upper bound on T which is more weaker but mo re easily to be treated. Lemma 3: T ≤ ( T ′ + D ′ ) D 2 d − 1 . Pr oof: Let x and y be an arbitrary pair of adjacent vertices in G S P with supports X and Y respectiv ely . Then d ( x , y ) ≤ d − 1 . Since they take di f fer values in p oints of ( X ∪ Y ) / ( X ∩ Y ) , d ( x , y ) ≥ | ( X ∪ Y ) / ( X ∩ Y ) | = | X | + | Y | − 2 | X ∩ Y | . Clearly , an binary ve cto r is uniquely determined by its sup port. Let x ′ , y ′ ∈ { 0 , 1 } n with supports X , Y respecti vely . T hen d ( x ′ , y ′ ) = | X | + | Y | − 2 | X ∩ Y | ≤ d ( x , y ) ≤ d − 1 . Furthermore, d ( x ′ , 0 ) = | X | = w t ( x ) = d ( x , 1 ) ≤ d − 1 , thereby x ′ ∈ G 2 , s imilarly , y ′ ∈ G 2 . Hence ( x ′ , y ′ ) ∈ E ( G 2 ) whenev er X 6 = Y . T herefore |{ ( X , Y ) : X , Y are supports of a pair of adjacent vertices in G S P with X 6 = Y } | ≤ | E ( G 2 ) | = T ′ , |{ ( X , Y ) : X , Y are supports of a pair of adjacent vertices in G S P with X = Y } | ≤ | V ( G 2 ) | = D ′ . Then |{ ( X , Y ) : X , Y are supports of a pair of adjacent vertices in G S P }| ≤ T ′ + D ′ , September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 10 which in conjuncti on with the fact |{ x : x ∈ G S P with support X }| ≤ D d − 1 completes the proof. QED. V u and W u [ ? ] have proved the following relation between T ′ and D ′ . Lemma 4: For e very cons tant 0 < α < 1 / 2 there is a positive constant ǫ such that the following holds: for d = αn , T ′ ≤ D ′ 2 − ǫ . Lemma 5: For any positive constant ǫ , 0 < α < 1 and any poly nomial function f ( x ) , there exists a posit iv e value N , for n ≥ N , f ( n ) ≤  n d − 1  ǫ , whenev er d = αn . Pr oof: It is well known that lim n →∞  n d  1 √ 2 nπ α (1 − α )  1 α α (1 − α ) 1 − α  n = 1 , then lim n →∞ f ( n )  n d − 1  ǫ = lim n →∞ f ( n )  n d  ǫ ·  n d  ǫ  n d − 1  ǫ = lim n →∞ f ( n )  p 2 nπ α (1 − α )( α α (1 − α ) 1 − α ) n  ǫ  n − d + 1 d  ǫ = 0 , which implies the statement. QED. Lemma 6: For e very cons tant 0 < α < 1 / 2 there is a positive constant ǫ such that the following holds: for d = αn , T ≤ D 2 D ′ ǫ . Pr oof: It follo ws from Lemma 3 th at T ≤ ( T ′ + D ′ ) D 2 d − 1 , and while it follo ws from the definitions of D and D ′ that D ≥  n d − 1  D d − 1 > D ′ D d − 1 /d . So we have D 2 T ≥ ( D ′ D d − 1 /d ) 2 ( T ′ + D ′ ) D 2 d − 1 = D ′ 2 d 2 ( T ′ + D ′ ) . Then by Lemma 4 there exists a positive constant ǫ such that D 2 T ≥ D ′ 2 d 2 ( D ′ 2 − ǫ + D ′ ) = D ′ ǫ d 2 (1 + D ′ ǫ − 1 ) ≥ D ′ ε 2 d 2 (14) September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 11 where ε = min( ǫ, 1) . By Lemma 5, there exists a pos itive constant N such t hat for n ≥ N , 2 d 2 = 2 α 2 n 2 < (  n d − 1  ) ε/ 2 < D ′ ε/ 2 , which in conjunction with (14) im plies D 2 T ≥ D ′ ε/ 2 . Since D 2 T > 1 always ho lds, there e xi sts a positiv e constant ε ′ such th at for 0 < n < N , D 2 T > D ′ ε ′ . T aking ǫ ′ = min( ε/ 2 , ε ′ ) , then for all d = α n , D 2 T ≥ D ′ ǫ ′ , namely T ≤ D 2 D ′ ǫ ′ . QED. Pr oof of Theor em2: W e are now ready to comp lete the proof of T heorem 2. Let α be a cons tant satisfying 0 < α < 1 / 2 . Then by the definitions of D and T , Theorem 4 and Lemma 6, for case d = αn there exists a positive constant ǫ such that α ( G P ) ≥ n ! 10 D  log 2 D − 1 / 2 log 2  D 2 3 D ′ ǫ  ≥ min( ǫ, 1) 20 · n ! D (log 2 D ′ + log 2 3) ≥ min( ǫ, 1) 20 · n ! V ( n,d − 1) log 2 V 2 ( n, d − 1) . Then we complete the proof. QED. Lemma 7: For e very constant 0 < α < 1 there is a positive const ant ǫ such that whenev er d = n α , T ≤ D 2 − ǫ . Pr oof: The proof relies on the following three lemmas. Lemma 8: For ever y constant 0 < α < 1 there is a positive const ant ǫ such that the following holds: for d = n α , T ′ ≤ D ′ 2 − ǫ . Pr oof: Let α ′ be a constant satisfying 0 < α ′ < 1 / 2 . Suppose the Hamming sphere graphs over { 0 , 1 } n defined for d = n α and d = α ′ n are G ′ S 2 and G ′′ S 2 respectiv ely . Let T ′ = | E ( G ′ S 2 ) | , T ′′ = | E ( G ′′ S 2 ) | and D ′ = | V ( G ′ S 2 ) | = | V ( G ′′ S 2 ) | = V 2 ( n, d − 1) − 1 . Clearly , there exists a positive integer N such that for n ≥ N , n α ≤ α ′ n . This im plies that for n ≥ N , E ( G ′ S 2 ) ⊆ E ( G ′′ S 2 ) , which means T ′ ≤ T ′′ . Then by lemm a 4, there exists a pos itive cons tant ǫ ′ such that T ′ ≤ T ′′ ≤ D ′ 2 − ǫ ′ , whenev er n ≥ N . Moreover , T ′ < D ′ 2 alwa ys holds, then there exists a posit iv e constant ǫ ′′ such that T ′ ≤ D ′ 2 − ǫ ′′ for 0 < n < N . T aking ǫ = min { ǫ ′ , ǫ ′′ } , then T ′ ≤ D ′ 2 − ǫ . QED. September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 12 Lemma 9: For every p air of constants 0 < α < 1 and 0 < δ < 1 sati sfying 1 − δ − α > 0 , whenev er d = n α , lim n →∞ D d − 1 D 1 − δ = 0 . Pr oof: By the definitions of D and D d − 1 we hav e lim n →∞ D d − 1 D 1 − δ ≤ lim n →∞ D d − 1 (( n d − 1 ) D d − 1 ) 1 − δ = lim n →∞ D δ d − 1 (( n d − 1 )) 1 − δ = lim n →∞ (( d − 1)! /e ) δ ( n ! ( d − 1)!( n − d +1) ) 1 − δ = lim n →∞ c ( d − 1)!( n − d +1) 1 − δ n ! 1 − δ (15) where constant c = e − δ . Then from Stirling ’ s formula lim n →∞ n ! √ 2 π n ( n e ) n = 1 it follows lim n →∞ D d − 1 D 1 − δ ≤ lim n →∞ c p 2 π ( d − 1)  d − 1 e  d − 1  p 2 π ( n − d + 1 )  n − d +1 e  n − d +1  1 − δ  √ 2 π n  n e  n  1 − δ = lim n →∞ c p 2 π ( d − 1)  n − d + 1 n  1 − δ 2  d − 1 e  d − 1  n − d +1 e  ( n − d +1)(1 − δ )  n e  n (1 − δ ) ≤ lim n →∞ c p 2 π ( d − 1)  d − 1 e  d − 1  n − d +1 e  ( n − d +1)(1 − δ )  n e  n (1 − δ ) (16) By multipl ying lim n →∞ e − 1 n α ( d − 1) ( d − 1) d − 1 = e − 1 lim n →∞  1 + 1 d − 1  d − 1 = e − 1 e = 1 and inequality n − d + 1 ≤ n , (16) yields lim n →∞ D d − 1 D 1 − δ ≤ lim n →∞ ce − 1 p 2 π ( d − 1)  n α e  d − 1  n e  ( n − d +1)(1 − δ )  n e  n (1 − δ ) = lim n →∞ ce − 1 p 2 π ( d − 1)  e δ n 1 − α − δ  d − 1 = 0 . QED. Lemma 10: For ever y cons tants 0 < α < 1 and ǫ > 0 , w hene ver d = n α , lim n →∞ d 2 + d D ǫ = 0 . September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 13 Pr oof: W e have lim n →∞ d 2 + d D ǫ = lim n →∞ n 2 α + n α D ǫ ≤ lim n →∞ n 2 α + n α  n d − 1  D d − 1  ǫ = lim n →∞ n 2 α + n α  n !( d − 1)! e ( d − 1)!( n − d +1)!  ǫ ≤ lim n →∞ n 2 α + n α  ( n − d +2) d − 1 e  ǫ = lim n →∞ ( n 2 α + n α ) e ǫ ( n − n α + 2) ( n α − 1) ǫ = 0 . QED. W e are now re ady to com plete the proof of Lemma 7. It follows from Lemma 8 that there is a positive constant ε satisfying T ′ ≤ D ′ 2 − ε . This combing with Lemma 3, we obtain T ≤ ( T ′ + D ′ ) D 2 d − 1 ≤ ( D ′ 2 − ε + D ′ ) D 2 d − 1 = ( D ′ D d − 1 ) 2 − ε D ε d − 1 + ( D ′ D d − 1 ) D d − 1 (17) It foll ows from Lemma 9 that for any constant 0 < δ < 1 − α , th ere exists a positive constant N , for n ≥ N satisfying D d − 1 < D 1 − δ , (18) and follows from the definition s o f D , D ′ , D d − 1 that D ′ D d − 1 ≤ d  n d − 1  D d − 1 ≤ dD . (19) By applications of (18) and (19) for (17), we have T ≤ ( dD ) 2 − ε D ε (1 − δ ) + d D D 1 − δ ≤ ( d 2 + d ) D 2 − εδ , (20) By Lemma 1 0 there exists a positive constant M , for n ≥ M satisfying d 2 + d ≤ D εδ / 2 . This in conju nction with (20) follo ws that for n ≥ max( N , M ) , T ≤ D 2 − εδ / 2 . Since T < D 2 alwa ys holds, t here exists a posi tiv e const ant ε ′ for 0 < n < max( N , M ) satisfying T ≤ D 2 − ε ′ . Therefore taking ǫ = min( εδ / 2 , ε ′ ) , for all n , T ≤ D 2 − ǫ . QED. September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 14 Pr oof of Theor em 3 W e are n ow re ady to complete the proo f of Theorem 3. Let α be a constant satisfying 0 < α < 1 . Then by the definitions of D and T , Theorem 4 and Lem ma 7, for case d = n α there exists an positive constant ǫ such that α ( G P ) ≥ n ! 10 D  log 2 D − 1 / 2 log 2  D 2 − ǫ 3  ≥ min( ǫ, 1) 20 · n ! D (log 2 D + log 2 3) ≥ min( ǫ, 1) 20 · n ! V ( n,d − 1) log 2 V ( n, d − 1) . Then we complete the proof. QED. I I I . I M P ROV E D T H E G I L B E R T - V A R S H A M OV B O U N D B Y C O N S I D E R I N G C OV E R E D B A L L S I N T E R S E C T I O N S A directly approach to im prove the Gilbert-V arshamov bound is to consider the intersections of the covered balls of the codew ords. By this approach, two bounds depend ed on other q uantities are given in th is section. Theor em 5: P ( n, d ) ≥ 2 · n ! V ( n, d − 1) + P [ n, d − 1 ] (21) Pr oof: Let C b e an ( n, P ( n, d ) , d ) P A. Let c ∈ C . Suppose a and b are two disti nct permut ations cove red by c on ly . Then it mus t hav e d ( a , b ) < d , ot herwise C ∪ { a , b } / { c } is an ( n, d ) P A of size P ( n, d ) + 1 , which is a contradiction. This implies there are at mo st P [ n, d − 1] permutation s cove red by c only . Then there are at least n ! − P ( n, d ) P [ n, d − 1] permutatio ns in S n cove red by at least 2 codewords. So we have P ( n, d ) V ( n, d − 1) = X c ∈ C | B ( x ) | ≥ n ! + |{ a ∈ S n : a is covered by at least t wo codewords. }| ≥ n ! + n ! − P ( n, d ) P [ n, d − 1] , which implies the claim of the theorem. QED. Clearly , P [ n, d − 1] ≤ V ( n, d − 1) , then the bound in Theorem 5 i s an improvement o ver the Gilbert-V arshamov bound for P A. While determini ng the exact values of P [ n, d − 1] s eems diffic ul t, for n bein g small v alu es, the u pper bounds on P [ n, d − 1] can be obtained by linear programming [ ? ], for general cases, boun ds on P [ n, d − 1] are give n below . September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 15 Pr opositi on 7: For all d ≤ n , P [ n, d − 1] ≥ max { ( d − 1)! , V ( n, ⌊ ( d − 1 ) / 2 ⌋ ) } , moreover for d being ev en, P [ n, d − 1] ≥ V ( n, d/ 2 − 1) +  n − 1 d/ 2 − 1  D d/ 2 . For all d ≤ n , P [ n, d − 1] ≤ max { L i, 0 + . . . + L i,i : i = ⌊ ( d − 1) / 2 ⌋ , . . . , d − 1 } , where L i,j is defined in Lemma 1. Pr oof: Clearly , the set of permut ations with s upports be subsets of { 0 , 1 , . . . , d − 2 } has pairwise distances less than d . This im plies P [ n, d − 1] ≥ | S d − 1 | = ( d − 1)! . And the set A = { x ∈ S n : w t ( x ) ≤ ⌊ ( d − 1) / 2 ⌋} has pairwise distances less than d also . This lead to P [ n, d − 1] ≥ | A | = V ( n, ⌊ ( d − 1) / 2 ⌋ ) . For case d being eve n , t he set B = { x ∈ S n : w t ( x ) = d/ 2 , x (0) 6 = 0 } has pairwise dist ances less than d , moreover the d istance from any permutation in A to any permutation in B is less than d . Hence For case d bein g ev en, P [ n, d − 1] ≥ | A | + | B | = V ( n, d / 2 − 1) +  n − 1 d/ 2 − 1  D d/ 2 . Suppose C is a su bset of S n with size of P [ n, d − 1] and pairwise di stances less than d . W ithout los s of generality , we assume that 1 is an element of C . If the maxim um weight of permutations in C i s i , then | C | ≤ L i, 0 + . . . + L i,i by Lemma 1. If i = ⌊ ( d − 1) / 2 ⌋ then C includes all the permutations with weights not more than ⌊ ( d − 1) / 2 ⌋ . T herefore we obtain t he upper bound on P [ n, d − 1] presented in the propositi on. QED. Remark: Anot her connection between P ( n, d ) and P [ n, d − 1] s hown in [ ? , Theorem 3] is that P ( n, d ) P [ n, d − 1] ≤ n ! . Theor em 6: Let C ′ be an ( n, M , d ) P A, then P ( n, d ) ≥ n ! M | B ( C ′ ) | . September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 16 Pr oof: Suppose C is an ( n, P ( n, d ) , d ) P A. Then for any x ∈ S n , ( x C / ( x C ∩ B ( C ′ ))) ∪ C ′ is an ( n, d ) P A with size | x C | − | x C ∩ B ( C ′ ) | + | C ′ | , where x C = { xc : c ∈ C } . Clearly , | x C | − | x C ∩ B ( C ′ ) | + | C ′ | ≤ P ( n, d ) . This in con junction with | x C | = | C | = P ( n, d ) and | C ′ | = M yields P ( n, d ) − | x C ∩ B ( C ′ ) | + M ≤ P ( n, d ) , i.e. | x C ∩ B ( C ′ ) | ≥ M . Then P x ∈ S n | x C ∩ B ( C ′ ) | ≥ M n ! . On the other hand, we have X x ∈ S n | x C ∩ B ( C ′ ) | = X b ∈ B ( C ′ ) X c ∈ C |{ x ∈ S n : xc = b }| = X b ∈ B ( C ′ ) X c ∈ C 1 = | B ( C ′ ) | P ( n, d ) Therefore M n ! ≤ | B ( C ′ ) | P ( n, d ) , in other words P ( n, d ) ≥ n ! M | B ( C ′ ) | . QED. Since n ! M | B ( C ′ ) | n ! V ( n,d − 1) = M · V ( n, d − 1) | B ( C ′ ) | = P c ∈ C ′ | B ( c ) | | ∪ c ∈ C ′ B ( c ) | , we can expect to improve the Gil bert-V arshamov bound on P ( n, d ) by constructing an ( n, d ) P A with relativ e small size of covered ball. For instance, in [ ? , Section 1, p.5 4], i t is suggested to choo se d permut ations wit h pairwis e dist ance exactly d . But ev aluation of | B ( C ′ ) | seems diffic ul t. I V . L OW E R B O U N D S F O R C E R TA I N C A S E S In this secti on, some new lower bounds for certain values of n and d are given. These new bounds follo w from two inequalities in P ( n, d ) which are deri ved by two cons tructions as follows, respectiv ely . Lemma 11: Suppose n ≥ d > 3 . Let Φ = { φ i } M i =1 be an ( n, M , d ) P A, and let ψ i : Z n − 1 7→ Z n − 1 be defined as follows ψ i ( x ) =    φ i ( x ) , for φ i ( x ) 6 = n − 1 φ i ( n − 1) , for φ i ( x ) = n − 1 . Then Ψ = { ψ i } M i =1 forms an ( n − 1 , M , d − 3) P A. Pr oof: Obviously , each ψ i ∈ Ψ is a permutation over Z n − 1 . For any 1 ≤ i, j ≤ M , i 6 = j , if φ i ( x ) 6 = n − 1 , φ j ( x ) 6 = n − 1 , t hen ψ i ( x ) 6 = ψ j ( x ) if and only if φ i ( x ) 6 = φ j ( x ) , thus we have d ( ψ i , ψ j ) ≥ |{ x : x ∈ Z n − 1 , φ i ( x ) 6 = n − 1 , φ j ( x ) 6 = n − 1 , φ i ( x ) 6 = φ i ( x ) }| ≥ d ( φ i , φ j ) − 3 ≥ d − 3 , September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 17 which implies the statement. QED. Lemma 12: Suppose n ≥ d > 2 . Let Φ be an ( n, M , d ) P A, and let Φ i = { φ ∈ Φ : φ ( i ) = n − 1 } , i = 0 , . . . , n − 1 . Suppose for s 6 = t and for any k 6 = s, t , | Φ s | ≥ | Φ t | ≥ | Φ k | , and Φ s = { φ s i } M 1 i =1 , Φ t = { φ t j } M 2 j =1 . Let ψ s i : Z n / { s } 7→ Z n / { s } and ψ t j : Z n / { s } 7→ Z n / { s } b e defined respectiv ely as follows ψ s i ( x ) = φ s i ( x ) , for x ∈ Z n / { s } , ψ t j ( x ) =    φ t j ( x ) , for x ∈ Z n / { s, t } φ t j ( s ) , for x = t. Then Ψ = { ψ s i } M 1 i =1 ∪ { ψ t j } M 2 j =1 is an ( n − 1 , d − 2) P A over Z n / { s } of size M 1 + M 2 ≥ 2 M n . Pr oof: Obviously , each ψ s i ∈ Ψ and each ψ t j ∈ Ψ are p ermutations over Z n / { s } . Moreover , for any permutations ψ s i , ψ t j ∈ Ψ and any x ∈ Z n / { s, t } , ψ s i ( x ) = φ s i ( x ) , ψ t j ( x ) = φ t j ( x ) . So for any di stinct perm utations ψ s i , ψ s j , ψ t i ′ , ψ t j ′ ∈ Ψ , d ( ψ s i , ψ s j ) ≥ d ( φ s i , φ s j ) − 2 ≥ d − 2 , d ( ψ t i ′ , ψ t j ′ ) ≥ d ( φ t i ′ , φ t j ′ ) − 2 ≥ d − 2 and d ( ψ s i , ψ t i ′ ) ≥ d ( φ s i , φ t i ′ ) − 2 ≥ d − 2 . Hence the l emma immediately follows. QED. From Lem ma 11 and Lemm a 12 we hav e the following theorem immedi ately . Theor em 7: For n ≥ d > 3 P ( n − 1 , d − 3) ≥ P ( n, d ) . (22) For n ≥ d > 2 P ( n − 1 , d − 2) ≥ 2 n P ( n, d ) . Cor ollary 2 : Let q be the power of prime number . Then P ( q , q − 4) ≥ ( q + 1) q ( q − 1) P ( q , q − 3) ≥ 2 q ( q − 1) P ( q − 1 , q − 4) ≥ ( q + 1)( q − 1) P ( q − 1 , q − 6) ≥ 2( q + 1)( q − 1) . Additionall y , P (1 1 , 5) ≥ 95040 and P (11 , 6) ≥ 15840 . Pr oof: If q is a prime-power , then it follows from Theorem 7 and Proposition 3 that P ( q , q − 4) ≥ P ( q + 1 , q − 1) = ( q + 1) q ( q − 1 ) (23) September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 18 T AB LE II C O M PA R I S O N O F L O W E R B O U N D S O N P ( q , q − 3 ) A N D P ( q , q − 4) Lowe r bound o n P ( q , d ) q is a prime po wer d Corollary 2 Proposition 4 Proposition 5 q ≡ 1 mod 6 , q 6 = 7 q − 3 2 q ( q − 1) q ( q − 1) q q ≡ 1 mod 6 and q ≡ 0 mod 5 q − 4 ( q + 1) q ( q − 1) q ( q − 1) 1 2 q 3 + 1 4 q 2 + 5 4 q q ≡ 1 mod 6 and q ≡ 1 mod 5 q − 4 ( q + 1) q ( q − 1) q ( q − 1) q q ≡ 1 mod 6 and q ≡ − 1 mo d 5 q − 4 ( q + 1) q ( q − 1) q ( q − 1) q 2 + q T AB LE III C O M PA R I S O N O F L OW E R B O U N D S O N P ( q − 1 , q − 4) Lowe r bound on P ( q − 1 , q − 4) q is a prime po wer Corollary 2 Proposition 4 Proposition 5 Proposition 6 and ( 6) and ( 6) and ( 5) q ≡ 1 mod 6 and q ≡ 0 mod 5 ( q + 1)( q − 1) q − 1 1 2 q 2 + 1 4 q + 5 4 ( q − 1)( θ ( q − 1) − 1) q ≡ 1 mod 6 and q ≡ 1 mod 5 ( q + 1)( q − 1) q − 1 1 ( q − 1)( θ ( q − 1) − 1) q ≡ 1 mod 6 and q ≡ − 1 mo d 5 ( q + 1)( q − 1) q − 1 q + 1 ( q − 1)( θ ( q − 1) − 1) and P ( q , q − 3) ≥ 2 q +1 P ( q + 1 , q − 1) = 2( q +1) q ( q − 1) q +1 = 2 q ( q − 1) . Moreover P (11 , 5) ≥ P (12 , 8) = 95040 , P (11 , 6) ≥ 2 12 P (12 , 8) = 2 · 95040 12 = 15840 . (6) in conjunction wit h (23 ), yi elds P ( q − 1 , q − 4) ≥ 1 q P ( q , q − 4) ≥ ( q + 1)( q − 1) . Additionall y , Theorem 7 in conj unction w ith (23), yields P ( q − 1 , q − 6) ≥ 2 q P ( q , q − 4) ≥ 2( q + 1)( q − 1) . QED. In general, for certain cases, the lower bounds g iv en by Corol lary 2 are m ore t ighter than the p re vi ous bounds, and they are compared i n T able II and III, where the functio n θ ( x ) in T able III is defined by (8). Moreov er , The ne w bounds P (11 , 5) ≥ 950 40 and P (11 , 6) ≥ 15840 are also tig hter than the pre vi ous bound P (11 , 5) ≥ 60940 and P (11 , 6) ≥ 9790 [ ? , T able 5, p.63] respectiv ely . mds September 7, 2021 DRAFT JOURNAL OF L A T E X CLASS FIL ES, V OL. 1, NO. 11, NO VE MBER 2002 19 November 18, 2002 September 7, 2021 DRAFT