Permutation Arrays Under the Chebyshev Distance

P erm utation Arra ys Under the Cheb yshev Distance ∗ T orleiv Kløv e, † T e-Tsung Lin, Shi-Ch un Tsai, and W en-Guey Tzeng ‡ Octob er 31, 2018 Abstract An ( n, d ) permutation arra y (P A) is a subset of S n with the p rop erty that the distance ( under some metric) b etw een any tw o p ermutations in the array is at least d . They b ecame p opular recently for comm u nication o ver pow er lines. Motiv ated by an application to flash memories, in this pap er the metric used is the Chebyshev metric. A num b er of different constructions are given as wel l as bou n ds on the size of suc h P A. 1 In tro duction Let S n denote the set of all p erm uta tions o f length n . A p erm utatio n array of length n is a subset of S n . Recen tly , Jiang et. al [1, 2] s ho wed an in teresting new application of p erm uta tio n ar rays for flash memories, where they used different distance metrics to inv estiga te efficient rewriting schemes. Under the multi- level fla sh memo ry model, we find the Chebyshev metric v er y a ppropriate for studying the recharging and error correcting issues. W e note it by d max . F o r π , σ ∈ S n , d max ( π , σ ) = max i | π i − σ i | . W e co nsider a noisy channel where pulse amplitude m o dulation (P AM) is used w ith different a mplitude levels for each p erm utatio n sy m b ol. The noise in the channel is a n indep endent Gaussian distribution with zero mean for each p osition. The r eceived sequence is the original permutation disto r ted by Gaus s ian noise, and its r a nking can be seen as a p ermut a tion, which can be different fro m the original o ne . T o study the co rrelations b etw een ranks, several metrics on p erm utatio ns were in tro duced, such as the Hamming dis ta nce, the minimum num be r of trans- po sitions taking one p ermutation to another, etc. [3], [4]. F or instance, Stoll ∗ The r esearc h wa s suppor ted in part b y the Nationa l Science Council of T aiw an under con tracts NSC-95-2221-E-009-094-MY3, NSC-96-2221-E-009-026, NSC-96-3114-P-001-002-Y and NSC-96-2219-E-009-013 and b y the Norwegian Research Council. † T. Kløve i s with the Department of Informatics, Univ ers i t y of B er gen, N-5020 Be r gen, Norwa y (Email : T orleiv.Klov e@ii. uib.no). ‡ T.-T. Lin, S.-C. Ts ai and W.-G. Tzeng are wi th the Departmen t of Computer Science, National Chi ao T ung Universit y , Hsi nc hu 30050, T aiwan (Email: atman.cs94g@nctu.edu.t w, sctsai@csie.nctu.edu.t w, wgtzeng@cs.nctu . edu.t w). 1 and Kurz [5] in vestigated a detection s c heme of per m utation arrays using Spear - man’s r ank correla tion. Chadwick and Kur z [6] studied the p ermutation a rrays based o n Kendall’s tau. Under the mode l of a dditiv e white Gaussia n noise (A W GN) [7], there is o nly a small probabilit y f o r any amplitude level to dev ia te significantly from the original one. This inspir ed us to use the C he byshev distance. Obser ve that t wo p ermutations with a lar g e Hamming dista nce ca n actually have a small Chebyshev distance and vice v ersa . They app ear to complemen t each other in some s ense. In this paper, we give a num b er of constructions of P A s . F or some w e give efficient deco ding alg orithms. W e a lso consider enco ding from vectors into per m utations. 2 Notations W e use [ n ] to denote the set { 1 , . . . , n } . S n denotes the set of all pe r m utations of [ n ]. F or any set X , X n denotes the set of all n -tuples with elements from X . Let ι denote the identit y p erm utatio n in S n . The Cheb yshev distance b e- t ween tw o p erm utations π , σ ∈ S n is d max ( π , σ ) = max { | π j − σ j | | 1 ≤ j ≤ n } . An ( n, d ) permutation array (P A) is a subset of S n with t he prop erty that the Chebyshev distance betw een any tw o distinct per m utations in the array is at leas t d . W e sometimes refer to the elements of a P A as co de words. The maxima l size of an ( n, d ) P A is deno ted by P ( n, d ). Let V ( n, d ) denote the num b er o f p ermut a tions in S n within Chebyshev distance d of the identit y per m utation. Since d max ( ι, σ ) = d max ( π , π σ ), the n umber of per m utations in S n within Chebyshev distance d of any pe rm utation π ∈ S n will also b e V ( n, d ). Bounds o n P ( n, a ) and V ( n, d ) will b e considered in Sec . 4. 3 Constructions In this section we give a num b er o f constr uctions of P As, one e x plicit and so me recursive. 3.1 An explicit c onstruction Let n and d b e given. Define C = { ( π 1 , . . . , π n ) ∈ S n | π i ≡ i (mo d d ) for all i ∈ [ n ] } . If n = ad + b , where 0 ≤ b < d , then C is an ( n, d ) P A and | C | = (( a + 1 )!) b ( a !) d − b . In pa rticular, we get the following b ound. 2 Theorem 1. If n = ad + b , wher e 0 ≤ b < d , then P ( n , d ) ≥ (( a + 1)!) b ( a !) d − b . Example 1. F or d = 2 , we get P ( 2 a, 2) ≥ ( a !) 2 . W e note that if 2 d > n , then a = 1 and b = n − d a nd so | C | = 2 n − d . If 2 d = n , then a = 2, b = 0 , a nd we hav e | C | = 2 d = 2 n − d as well. How ever, if 2 d < n , then | C | > 2 n − d . Esp ecially , when d is small r elativ e to n , | C | is muc h larger than 2 n − d . F or example, for n = 30 , d = 2, | C | / 2 n − d ≈ 6 . 37 × 10 15 . This cons truction has a very simple deco ding algorithm. F or d ≥ 2 t + 1, we can co rrect er ror up to size t in a n y co or dinate. F or co ordinate i , the co deword has v alue π i ≡ i (mo d d ). Suppo se that this co ordina te is c ha ng ed into σ = π i + u , w he r e | u | ≤ t . Then π i is the int eg er congruent to i which is closest to σ . Therefor e, deco ding of p o sition i is done b y first computing a ≡ i − σ (mo d d ) , where − ( d − 1) / 2 ≤ a ≤ ( d − 1) / 2 . Then a = − u , and so we deco de into σ + a = π i . 3.2 First recursiv e construction Let C be a n ( n, d ) P A o f s ize M , and let r ≥ 2 b e an integer. W e define an ( rn, r d ) P A , C r , of size M r as follows: for each multi-set of r co de w o rds from C , ( π ( j ) 1 , . . . , π ( j ) n ) , j = 0 , 1 , . . . , r − 1 , let ρ j = ( r π ( j ) 1 − j, . . . , r π ( j ) n − j ) , j = 0 , 1 , . . . , r − 1 , and include ( ρ 0 | ρ 1 | . . . | ρ r − 1 ) as a co deword in C r . It is clear that under this construction the d is tance b et ween a n y tw o distinct ρ j , ρ j ′ is a t least rd . It is also eas y to chec k that ( ρ 0 | ρ 1 | . . . | ρ r − 1 ) ∈ S r n . Hence | C r | = P ( n, d ) r . In particular, we get the following b ound. Theorem 2. If n > d and r ≥ 2 , then P ( r n, r d ) ≥ P ( n , d ) r . 3.3 Second r ecursiv e construction F or a p ermut a tion π = ( π 1 , π 2 , . . . , π n ) ∈ S n and an integer m , 1 ≤ m ≤ n + 1 define ϕ m ( π ) = ( m, π ′ 1 , π ′ 2 , . . . , π ′ n ) ∈ S n +1 3 by π ′ i = π i if π i ≤ m, π ′ i = π i + 1 if π i > m. Let C b e a n ( n, d ) P A, and let 1 ≤ s 1 < s 2 < · · · < s t ≤ n + 1 be integers. Define C [ s 1 , s 2 , . . . , s t ] = { ϕ s j ( π ) | 1 ≤ j ≤ t, π ∈ C } . Theorem 3. If C is an ( n, d ) P A of size M and s j + d ≤ s j +1 for 1 ≤ j ≤ t − 1 , then C [ s 1 , s 2 , . . . , s t ] is an ( n + 1 , d ) P A of size tM . Theorem 4. If C is a n ( n, d ) P A of size M and n ≤ 2 d , then C [ d ] is an ( n + 1 , d + 1) P A of size M . Pro of. If j > j ′ , then d max ( ϕ s j ( π ) , ϕ s j ′ ( σ )) ≥ s j − s j ′ ≥ d. Next, conside r j ′ = j . If π , σ ∈ C , π 6 = σ , then w.l.o .g, there exist an i such that π i ≥ σ i + d . Hence d max ( ϕ s j ( π ) , ϕ s j ( σ )) ≥  π i − σ i + 1 > d if π i > s j ≥ σ i , π i − σ i ≥ d otherwise. This proves Theor em 3. T o complete the pro of of Theorem 4 we note that π i ≥ σ i + d ≥ d + 1 > d, and σ i ≤ π i − d ≤ n − d ≤ d. Hence π i > d ≥ σ i and so d max ( ϕ s j ( π ) , ϕ s j ( σ )) ≥ d + 1 . The co nstructions imply b ounds on P ( n , d ). First, choos ing t = ⌊ n/ d ⌋ + 1, s t = n + 1 and s j = ( j − 1 ) ⌊ n/d ⌋ + 1 for 1 ≤ j ≤ t − 1, we get the following bo und. Theorem 5. If n > d ≥ 1 , t hen P ( n + 1 , d ) ≥ j n d k + 1  P ( n , d ) . 4 Example 2. In Example 1 we showe d that the explicit c onst r u ction implie d that P ( 2 a, 2) ≥ ( a !) 2 . Combining The or em 5 and se ar ch, we c an impr ove this b oun d. We have found t hat P (7 , 2) ≥ 582 , se e the t able at the end of t he next se ct ion. F r o m r ep e ate d u se of The or em 5 we get P ( 2 a, 2) ≥ ( a ( a − 1) · · · 5) 2 · 4 P (7 , 2) ≥ 97 24 ( a !) 2 . Theorem 4 implies the following b ound. Theorem 6. If d < n ≤ 2 d , then P ( n + 1 , d + 1 ) ≥ P ( n, d ) . Theorem 6 shows in particula r tha t for a fixed r , P ( d + 1 + r, d + 1 ) ≥ P ( d + r, d ) for d ≥ r. (1) W e will show that P ( d + r, d ) is b ounded. W e show the following theor em. Theorem 7. F or fixe d r , ther e ex ist c onstants c r and d r such that P ( d + r , d ) = c r for d ≥ d r . Mor e over, c r ≤ 2 2 r (2 r )! (2) and d r ≤ 1 + (2 r − 1 ) c r − r . (3) Remark. The main p o in t o f Theor em 7 is the existence of c r and d r . The actual b ounds g iv en are probably quite weak in genera l. F o r example, Theorem 7 gives the b ounds c 1 ≤ 8 a nd d 1 ≤ 8 . In Theorem 8 b elo w, we will show that c 1 = 3 a nd d 1 = 2. Theorem 7 gives c 2 ≤ 38 4 and d 2 ≤ 1151, whereas n umer ical computation indica te that c 2 = 9 and d 2 = 5. W e split the pro of of Theorem 7 into three lemma. Lemma 1. If d ≥ r , then P ( d + r, d ) ≤ 2 2 r (2 r )! . Pro of. Suppose that there ex ists a n ( d + r, d ) P A C o f size M > 2 2 r (2 r )!. W e call the integers 1 , 2 , . . . , r and d + 1 , d + 2 , . . . , d + r p otent , the first r smal ler p otent , the la s t r lar ger p otent . Two p oten t integers are ca lle d e quip otent if b oth a re smaller p otent or b oth are lar ger p otent . If the distance betw e e n tw o p ermutations ( π 1 , π 2 , . . . , π n ), ( ρ 1 , ρ 2 , . . . , ρ n ) is at least d , then there exists some p osition i such that, w.l.o.g , π i − ρ i ≥ d , Then π i is a la rger p oten t element and ρ i is s ma ller p otent. Each per m utation in S d + r contains 2 r p otent elements and we ca ll the set of p ositions of these the p otency supp ort χ ( π ) of the p ermutation, that is, the p otency supp ort of π is χ ( π ) = { i | 1 ≤ π i ≤ r } ∪ { i | d + 1 ≤ π i ≤ d + r } . 5 The p otency supp ort of C is the union of the p otency suppo r t of the p ermuta- tions in C , that is χ ( C ) = { i | 1 ≤ π i ≤ r for some π ∈ C } ∪ { i | d + 1 ≤ π i ≤ d + r for some π ∈ C } . Let π ∈ C . F or each ρ ∈ C , ρ 6 = π , we hav e d ( π , ρ ) ≥ d . Hence there exists some i ∈ χ ( π ) s uc h that ρ i is p otent. Therefore, the s et { ( ρ, i ) | ρ ∈ C and i ∈ χ ( π ) } contains at least 2 r + ( M − 1) > M elements. Hence there is an i ∈ χ ( π ) such that |{ ρ ∈ C | ρ i is po ten t }| > M / (2 r ) > 2 2 r (2 r − 1 )! . Since { ρ ∈ C | ρ i is po ten t } = { ρ ∈ C | ρ i is smalle r p oten t } ∪{ ρ ∈ C | ρ i is larger p otent } , there exis ts a subset C 1 ⊂ C suc h that | C 1 | > 2 2 r − 1 (2 r − 1 )! and the elements in po s ition i 1 = i ar e equip otent. W e ca n now re p eat the pr ocedur e. Let π ∈ C 1 . Ther e must exist an i 2 ∈ χ ( π ) \ { i 1 } such that |{ ρ ∈ C 1 | ρ i 2 is po ten t }| ≥ | C 1 | / (2 r − 1 ) > 2 2 r − 1 (2 r − 2 )! . Hence we ge t subset C 2 ⊂ C 1 such that | C 2 | > 2 2 r − 2 (2 r − 2 )! and the ele ments in po sition i 2 are equipo ten t (and the elements in p osition i 1 are equip otent). Repe a ted use of the same argument will pr o duce for ea c h j , 1 ≤ j ≤ 2 r a set C j such that | C j | > 2 2 r − j (2 r − j )! and for j p ositions i 1 , i 2 , . . . i j , the elements in thos e p ositions are all equip otent . In particular , | C 2 r | > 1, all p ermutations in C 2 r hav e the sa me p otency supp ort { i 1 , i 2 , . . . , i 2 r } , and for each o f these p ositions, all the element s in that p osition are eq uip otent. This is a co n tradictio n since the dista nce b et ween tw o such per m utations must b e less than d . Hence the assumption that a P A of size larger than 2 2 r (2 r )! exists le a ds to a cont r adiction. Lemma 1 combined with (1) proves the existence of c r and d r and gives the bo und (2). 6 Lemma 2. If C is a ( d + r, d ) P A of size M wher e d > r and d + r > | χ ( C ) | , then ther e exists a ( d − 1+ r , d − 1) P A of size M . In p articular, if M = P ( d + r, d ) , then P ( d − 1 + r, d − 1 ) = P ( d + r , d ) . Pro of. Repla ce all elements in ra nge r + 1 , r + 2 , . . . , d in the p ermu ta tions of C b y a star ∗ which will deno te ”unsp ecified”. The per m utations in C is transformed into ve ctors containing the p oten t elements and d − r stars . Note that if we replac e the unsp ecified elements in each vector by the integers r + 1 , r + 2 , . . . , d in so me o rder, we get a p ermutation, and the distance b et ween t wo such p erm utatio ns will b e at least d since we hav e no t changed the p oten t elements. Since the leng th d + r of C is larg er than | χ ( C ) | , there exists a po sition where all the v ec to rs contains a star . Remov e this p osition from ea c h vector and r educe all the lar ger p oten t elements by one. This given a se t o f M vectors of length d − 1 + r and s uc h that the distance b e tw e e n any tw o is at least d − 1 . Replacing the d − 1 − r stars in each vector by r + 1 , r + 1 , . . . , d − 1 in some order, we get an ( d − 1 + r , d − 1) P A of size M . If M = P ( d + r, d ), then we get P ( d − 1 + r, d − 1 ) ≥ P ( d + r , d ) . Since P ( d − 1 + r, d − 1 ) ≤ P ( d + r , d ) by (1), the lemma follows. Lemma 3. If C is a ( d + r, d ) P A of size M and d ≥ r , then | χ ( C ) | ≤ M (2 r − 1) + 1 . Pro of. Each p ermutation has po tency supp ort of size 2 r . The po tency suppo rt of any tw o p ermutations in C m ust o verlap since their distance is at least d . Hence each p erm utatio n after the first will contribute at mo st 2 r − 1 new elements to the total po tency supp ort. Ther efore, | χ ( C ) | ≤ 2 r + ( M − 1)(2 r − 1) . Remark. By a more in volved analysis , we can improv e this bound somewhat. F or ex ample, we see that tw o new p ermutations ca n contribute a t most 4 r − 3 to the total supp ort. W e ca n now co mplete the pro of o f Theorem 7 . Let C b e a ( d + r , r ) co de of size c r . By Lemma 3, | χ ( C ) | ≤ c r (2 r − 1) + 1. If d > 1 + c r (2 r − 1) − r , then d + r > | χ ( C ) | . Hence, by L e mma 2, P ( d − 1 + r, d − 1) = P ( d + r , d ). Therefore, d r ≤ 1 + c r (2 r − 1) − r , that is , (3) is s atisfied. This co mpletes the pro of of Theorem 7 . 7 Theorem 8. We have P ( d + 1 , d ) = 3 for d ≥ 2 . Pro of. W e use the same nota tion as in the pro of of Lemma 2. Le t C b e an ( d + 1 , d ) P A. The o nly p otent elements are 1 a nd n . W.lo.g. we may as s ume the first p erm utatio n in C is (1 , n, ∗ , ∗ , . . . ) wher e ∗ denotes so me unsp ecified int eg er in the r ange 2 , 3 , . . . , d . W.l.o.g, a s econd p ermutation ha s o ne of three forms: ( n, 1 , ∗ , ∗ , . . . ) , ( n, ∗ , 1 , ∗ , . . . ) , ( ∗ , 1 , n , ∗ , . . . ) . W e see that if the second p ermutation is of the first for m, there cannot b e mor e per m utations. If the second p ermutation is of the form ( n, ∗ , 1 , ∗ , . . . ), then there is only one p ossible form for a thir d p ermutation, namely (1 , ∗ , n, ∗ , . . . ). Hence we see that P ( d + 1 , d ) ≤ 3 a nd that P ( d + 1 , d ) = 3 for d ≥ 2 . T o determine P ( d + r, d ) along the same lines fo r r ≥ 2 seems to be difficult bec ause of the many cases that have to b e considered. Even to determine P ( d + 2 , d ) will inv o lv e a la rge num b er of cases . F or example for the second per m utation there ar e 138 esse ntially different po ssibilities for the four p ositions in the po tency supp ort of the firs t p ermutation. F o r each o f these there are man y po ssible third p ermu ta tions, etc. 3.4 Enco ding/deco ding of some P A constructed b y the second r ecursiv e construction Suppo se we start with the P A C d = { (1 , 2 , 3 , . . . , d ) } . F or ν = d, d + 1 , . . . , n − 1 let C ν +1 = C ν [1 , ν + 1] . Then C n is a n ( n, d ) P A of size 2 n − d . F or some applications, w e may wan t to map a se t of binary v ec tors to a p erm uta tio n ar ray . One algorithm for mapping a binary v ecto r ( x 1 , x 2 , . . . , x n − d ) into C n would be to use the rec ursive construction of C n by mapping ( x 1 , x 2 , . . . , x i ) into a per m utation π in C d + i . Recursively , we can then map ( x 1 , x 2 , . . . , x i , 0) to f 1 ( π ) and ( x 1 , x 2 , . . . , x i , 1) to f d + i +1 ( π ). How ever, there is an alternative alg orithm which requires less work. Retrac- ing the steps of the constr uction, we see that given s ome initial part of length less than n − d of a per m utation in C n , there are e xactly tw o p ossibilities fo r the next elemen t, one ”larg er” and one ”smaller ” . Mo re precise ly , induction shows that if the initial part of length i − 1 contains exactly t ” smaller” elements, then element num b er i is either t + 1 (the ”smaller ”) or n − i + t + 1 (the ”large r ”). This is the ba sis for a simple ma pping from Z n − d 2 to C n . W e give this algo rithm in Fig ure 1. W e see that the difference b etw een the larger and the smaller ele ment in po sition i ≤ n − d is n − i . Hence we can recover from any err or o f size less than 8 Input: ( x 1 , . . . , x n − d ) ∈ Z n − d 2 Output: ( π 1 , . . . , π n ) ∈ C n for i ← n − d + 1 to n do x i ← 0 ; t ← 0; //* t is the num b er of zeros seen so far.*/ / for i ← 1 to n do if x i = 0 then { π i ← t + 1; t ← t + 1 ; } else { π i ← n − i + t + 1; } Figure 1 : Algorithm mapping Z n − d 2 to C n Input: ( π 1 , . . . , π n ) ∈ [ n ] n Output: ( x 1 , . . . , x n − d ) t ← 0; //* t is num ber o f zeros determined. *// for i ← 1 to n − d do if π i < ( n − i ) / 2 + t + 1 then { x i ← 0 ; t ← t + 1 ; } else { x i ← 1 ; } Figure 2: Deco ding algo rithm recovering the bina ry preimag e from a corrupted per m utation in C n . 9 ( n − i ) / 2 by choosing the clo sest o f the t wo po ssible v alues, a nd the corresp onding binary v alue. W e give the deco ding alg orithm in Fig ure 2. Without go ing into all details, we s ee that we ca n ge t a similar mapping from q -a ry vectors. Now we star t with the P A C ( q − 1) d = { (1 , 2 , 3 , . . . , ( q − 1) d ) } . F or ( q − 1) d ≤ ν ≤ n − 1 let s j = ( j − 1) ⌊ ν / ( q − 1) ⌋ + 1 for 1 ≤ j ≤ q − 1 and s q = ν + 1. Let C ν +1 = C ν [ s 1 , s 2 , . . . , s q ] . Then C n is a n ( n, d ) P A o f size q n − ( q − 1) d . Encoding and deco ding cor recting error s of size at most ( d − 1 ) / 2, based on the recur sion, is again relatively simple. 4 F urther b ou nds on P ( n, d ) 4.1 General b ounds Since d max ( π , σ ) ≤ n − 1 for any tw o distinct p ermutations in S n , we hav e P ( n, n ) = 1. Therefor e, we o nly consider d < n . Since the spheres of r adius d in S n all hav e size V ( n, d ), we can get a Gilber t t y p e low er b ound on P ( n, d ). Theorem 9. F or n > d ≥ 2 we have P ( n, d ) ≥ n ! V ( n, d − 1) . Pro of. It is clear that the following greedy algor ithm pr oduces a p ermutation array with car dina lit y at leas t n ! /V ( n, d − 1). 1. Start with a ny p ermutation in S n . 2. Cho o se a p ermutation whose distance is at least d to all previo us chosen per m utations. 3. Repea t step 2 as lo ng as such a p ermut a tion exis ts. Let C b e the p ermutation ar ray pro duced by the ab o ve gre e dy algo rithm. O nce the alg orithm stops, S n will be cov er ed by the | C | s pher es of r adius d − 1 centered at the co de w o rds in C . Thus n ! ≤ | P | · V ( n, d − 1) whic h implies our claim. Similarly , since the sphere s V ( n, ⌊ ( d − 1) / 2 ⌋ ) a re disjoint, we get the following Hamming type upp er bo und. Theorem 10. If n > d ≥ 1 , t hen P ( n, d ) ≤ n ! V ( n, ⌊ ( d − 1) / 2 ⌋ ) . 10 If n ≤ 2 d and d is even, we can com bine the b ound in Theore m 10 with Theorem 6 to get the follo wing b ound whic h is stro nger than the o rdinary Hamming b ound, at least in the cases we have tested. Theorem 11. If d is even and 2 d ≥ n > d ≥ 2 , then P ( n, d ) ≤ ( n + 1)! V ( n + 1 , d/ 2) . Example 3. F or n = 11 and d = 6 , The or em 10 gives P (11 , 6) ≤  11! V (11 , 2)  =  11! 11854  = 33 67 wher e as The or em 11 gives P (11 , 6) ≤  12! V (12 , 3)  =  12! 56317 2  = 85 0 . Remark. W e can of course use Theorem 6 rep eatedly r times and then Theorem 10 to get P ( n, d ) ≤ ( n + r )! V ( n + r, ⌊ ( d + r − 1) / 2 ⌋ ) for all r ≥ 0 . How ever, it app ears we get the b est b ounds for r = 1 when d is even a nd r = 0 when d is o dd. In genera l, no simple e xpression of V ( n, d ) is known. A survey of known results as well as a num b er of new results o n V ( n, d ) were given by Klø v e [8 ]. Here we briefly give some main r esults. As obs erved by Lehmer [9], V ( n, d ) ca n be expr essed as a p ermanent. The per manen t of an n × n matrix A is defined by per A = X π ∈ S n a 1 ,π 1 · · · a n,π n . In pa rticular, if A is a (0 , 1)- ma trix, then per A = |{ π ∈ S n : a i,π i = 1 for all i }| . Let A ( n,d ) be the n × n matrix with a ( n,d ) i,j = 1 if | i − j | ≤ d and a ( n,d ) i,j = 0 otherwise. Lemma 4. V ( n, d ) = p er A ( n,d ) . Pro of. V ( n, d ) = |{ π ∈ S n : d max ( ι, π ) ≤ d }| = |{ π ∈ S n : | i − π i | ≤ d for a ll i } | = |{ π ∈ S n : a ( n,d ) i,π i = 1 for all i }| = p er A ( n,d ) . 11 F or fixed d , V ( n, d ) satisfies a linear recurre nce in n . A pr oof is given in [1 0] (Prop osition 4.7 .8 on page 24 6). F o r 1 ≤ d ≤ 3 these recurr ences were deter - mined explicitly by Lehmer [9], and for 4 ≤ d ≤ 6 by Kløve [8]. In par ticular, this implies that lim n →∞ V ( n, d ) 1 /n = µ d , where µ d is the lar gest ro ot of the minimal p olynomial cor resp onding to the linear recurr ence of V ( n, d ). Lehmer [9 ] deter mined µ d approximately for d = 1 , 2 , 3 and Klø v e [8 ] for d ≤ 8. F or an n × n (0 , 1)-matrix it is known (see Theorem 1 1.5 in [11]) that per A ≤ n Y i =1 ( r i !) 1 /r i , where r i is the num b er of ones in row i . F or A ( n,d ) we clea rly hav e r i ≤ 2 d + 1 for all i . Hence V ( n, d ) ≤ [(2 d + 1)!] n/ (2 d +1) for a ll n (4) and µ d ≤ [(2 d + 1 )!] 1 / (2 d +1) . In T able 1 we give µ d and this upp er bo und. T able 1: µ d and its upp er bo und. d µ d [(2 d + 1 )!] 1 / (2 d +1) µ d / (2 d + 1) 1 1 . 61 803 1 . 8171 2 0 . 5393 4 2 2 . 33 355 2 . 6051 7 0 . 4667 1 3 3 . 06 177 3 . 3800 2 0 . 4373 9 4 3 . 79 352 4 . 1471 7 0 . 4215 0 5 4 . 52 677 4 . 9092 4 0 . 4115 2 6 5 . 26 082 5 . 6676 9 0 . 4046 8 7 5 . 99 534 6 . 4234 2 0 . 3996 9 8 6 . 73 016 7 . 1770 4 0 . 3958 9 W e note that for large d , µ d / (2 d + 1) ≈ 1 /e . Combining Theo rem 9 and (4) we get Corollary 1. F or n > d ≥ 1 , we have P ( n, d ) ≥ n ! [(2 d − 1 )!] n/ (2 d − 1) . 12 4.2 T able of b ounds on P ( n, d ) W e hav e us ed the following greedy algorithm to find a n ( n, d ) P A C : Let the ident ity p erm utatio n in S n be the fir s t p erm utatio n in C . F o r any set of p er- m uta tio ns chosen, choo se as the next pe r m utation in C the lexico g raphically next p erm uta tion in S n with distance at least d to the chosen p ermutations in C if such a p ermutation exists. The size of the resulting P A is of c ourse a low er bo und on P ( n, d ). The low er b ounds in T able 2 were in most cases found by this greedy a lg o- rithm. F or n = 8, d = 5, the gr eedy a lgorithm gave a P A of size 26. How ever, P (8 , 5) ≥ P (7 , 4) ≥ 28 by Theo r em 6. Similar ly , P (10 , 7) ≥ P (9 , 6) ≥ P (8 , 5 ) ≥ 2 8 . Some other of the low er b ounds ar e a lso determined using Theorem 6. They are mar k ed by ∗ . The upp er bo und is the Hamming t y pe b ound in Theo rem 10 or it’s mo dified b ound in Theo rem 11. Since P ( n, 1) = n ! for all n , this is not included in the table. T able 2: Bounds on P ( n, d ). d = 2 d = 3 d = 4 n = d + 1 3 3 3 n = d + 2 6 − 24 9 9 − 1 2 n = d + 3 29 − 1 20 20 − 3 4 28 − 43 n = d + 4 90 − 7 20 84 − 148 6 8 − 166 n = d + 5 582 − 5040 401 − 733 28 3 − 407 7 d = 5 d = 6 d = 7 n = d + 1 3 3 3 n = d + 2 9 − 12 9 − 18 9 − 18 n = d + 3 28 ∗ − 43 28 ∗ − 60 2 8 ∗ − 60 n = d + 4 95 − 166 95 ∗ − 216 95 ∗ − 216 n = d + 5 2 36 − 71 4 236 ∗ − 850 236 ∗ − 850 5 Conclusion W e g ive a num b er o f co nstructions of p ermutations ar r a y s under the Chebyshev distance, some with efficient deco ding algor ithms. W e also consider an explicit mapping of vectors to p ermutations with efficient enco ding/deco ding. Finally , we give so me b o unds on the size of P A s under the Chebyshev distance. 13 References [1] A. Jiang , R. Mateescu, M. Sch wartz a nd J. Bruck, “Ra nk Mo dulation for Flash Memor ies,” in Pr o c. IEEE Internat. S ymp. on Inform. Th. , 200 8 , pp. 1731- 1735. [2] A. Jiang , M. Sch wartz and J. 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