Table Of Content

Decoding Frequency Permutation Arrays under Infinite norm Min-Zheng Shieh Shi-Chun Tsai Department of Computer Science Department of Computer Science National Chiao Tung University National Chiao Tung University 1001 University Road, Hsinchu, Taiwan 1001 University Road, Hsinchu, Taiwan Email: [email protected] Email: [email protected] 9 0 Abstract—Afrequencypermutationarray(FPA)oflengthn= and associate each symbol i ∈ {1,...,m} with a frequency 0 2 mmλsyamnbdodlsi,stwanhceeredeaischassyemtbooflpaeprpmeuartastieoxnasctolynλatmimuelstisaentdovthere fi. Thenthe message is transmittedas a series of corresponding frequencies. For example, to send a message encoded n distance between any two elements in the array is at least d. a FPA generalizes the notion of permutation array. In this paper, as (1,2,2,1,3,3), we can transmit the frequency sequence J under the distance metric ℓ∞-norm, we first prove lower and (f1,f2,f2,f1,f3,f3) one by one. 4 upperboundsonthesizeofFPA.Thenwegiveaconstructionof 1 FPAwithefficientencodinganddecodingcapabilities.Moreover, For flash memory application, different from the approach we show our design is locally decodable, i.e., we can decode a byJianget.al.[5],[6],wecanuseFPA toprovidemulti-level ] message bit by reading at most λ+1 symbols, which has an flash memory with error correcting capability. For example, T interesting application for private information retrieval. suppose a multi-level flash memory, where each cell has m I s. I. INTRODUCTION states, which can be changedby injecting or removingcharge c Let n,m and λ be positive integers with n = mλ, into or from it. Over injecting or charge leakage will alter [ and Sλ be the set of all permutations on the multiset the state as well. We can use the charge ranks of n cells 1 λ n λ to represent a permutation from Snλ, i.e., the cells with the v lowestλchargelevelsrepresentsymbol1,andsoon.Withour {1,...,1,...,m,...,m}. A frequency permutation array 71 (FzPA})| is{ a suzbset}|of {Snλ for some positive integers m, λ ceaffincbieentuseendcoindiflnagshanmdedmeocroydisnygsteamlgotroitrhemprse,saen(tλi,nnfo,rdm)-aFtiPoAn 9 and n = mλ. A (λ,n,d)-FPA is a subset of Snλ and and correct errors caused by charge level fluctuation. 1 the distance between any pair of distinct permutations is at . least d under any metric, such as Hamming distance, ℓ - 1 ∞ A locally decodable code has an extremelyefficient decod- 0 norm, etc. Permutation array (PA) is simply a special case ing for any message bit by reading at most a fixed number 9 of FPA by choosing λ = 1. With a fixed length n, FPA of symbols from the received word. Suppose that a FPA is 0 has a smaller set of symbols than PA. Thus, codes with FPA applied to a multi-level flash memory where the length of : v have a better information rate than those with PA. A widely a codeword is nearly a block of cells (about 105)[1]. This i adopted approach to building PAs under Hamming distance, X feature allows us to retrieve the desire message bits from a see for example [2], is using distance preserving mappings multi-level flash without accessing the whole block. With the r a or distance increasing mappings from Z2k to Sn1. Most of locally decodableproperty,we can raise the robustness of the those encodingschemesareefficientbutit is notclear how to code without loss of efficiency. On the other hand, locally decode efficiently. Lin et al. [8] proposed a couple of novel decodablecodeshavebeenunderstudyforyears,see [10] for constructionswith efficientencodinganddecodingalgorithms a survey and [15], [3] for recent progress. They are related to for PAs under l -norm. FPA was proposed by Huczynska ∞ a cryptographic protocol called private information retrieval and Mullen [4] as a generalization of PA. They gave several (PIRforshort).We show ourconstructionof FPA canalso be constructionsofFPAunderHammingdistanceandboundsfor used in cryptographic application. themaximumarraysize.Inthispaper,weextendstheideasin [8] to constructing FPA under l -norm. We prove lower and Notations:Letmandλbepositiveintegersandletn=mλ ∞ upper bounds of FPA. Then we show the efficient encoding throughout the paper unless stated otherwise. We use [n] to anddecodingalgorithms.Besides, we show thatourFPAs are representtheset{1,...,n}.Sλ denotesthepermutationsover n locally decodable codes under l -norm. λ λ ∞ Recently,researchershavefoundthatPAshaveapplications the multiset {1,...,1,...,m,...,m}. For two vectorsx and inareassuchaspowerlinecommunication(e.g.[9],[12],[13] y of the samezdi}m|en{sion, lzet l∞}|(x,{y)=maxi|xi−yi|. We and[14]),multi-levelflashmemories(see[5]and[6]).Similar say two permutationsx andy are d-close to eachotherunder to the application of PAs on power line communication, we metric δ(·,·) if δ(x,y) ≤ d. The identity permutation Iλ in n can encode a message as a frequency permutation from Sλ Sλ is (1,...,1,...,m,...,m). n n II. LOWERANDUPPERBOUNDS Itimpliesthatrow(ℓλ−λ+1)throughrowℓλofA(λ,λm,d) are identicaland so are columnsindexedfrom (ℓλ−λ+1)to ℓλ Let F (λ,n,d) be the cardinality of the maximum ∞ for every ℓ∈[m]. Thus, we have A(λ,λm,d) =A(1,m,d)⊗1 (λ,n,d)-FPA and V (λ,n,d) be the number of elements in λ Sλ being d-close to∞the identity Iλ under ℓ -norm. In this where ⊗ is the operator of tensor product and 1λ is a λ×λ n n ∞ matrix with all entries equal to 1. For example, take λ = 2, section, we give a Gilbert type lower bound and a sphere m=5 and d=2: packingupperboundofF (λ,n,d)byboundingV (λ,n,d). ∞ ∞ First, we show that any d-radius ball in Sλ under l -norm 1 1 1 0 0 n ∞ has the same cardinality.  1 1 1 1 0  1 1 Claim 1: Foranyx=(x ,...,x )∈Sλ,thereareexactly A(1,5,2) = 1 1 1 1 1 ,1 = 1 n n   2 (cid:18) 1 1 (cid:19) V∞(λ,n,d) y’s in Snλ such that l∞(x,y)≤d.  0 1 1 1 1  Proof: Since every i ∈ [m] appears exactly λ times in  0 0 1 1 1    x, there exists a permutation π ∈ S1 such that x = π◦Iλ. n n As a consequence, we have that l (Iλ,z) = l (x,π ◦z) 1 1 1 1 1 1 0 0 0 0 ∞ n ∞ for any z ∈ Sλ. Let Z = {z : z ∈ Sλ,l (Iλ,z) ≤ d},  1 1 1 1 1 1 0 0 0 0  n n ∞ n Y ={π◦z :z ∈Z}andY¯ =Sλ−Y.Foranyy ∈Y,wehave 1 1 1 1 1 1 1 1 0 0 n   l (x,y) = l (Iλ,π−1◦y) ≤ d, since π−1◦y ∈ Z. While  1 1 1 1 1 1 1 1 0 0  ∞ ∞ n   for y′ ∈ Y¯, l∞(x,y′) = l∞(Inλ,π−1 ◦y′) > d. Therefore, A(2,10,2) = 1 1 1 1 1 1 1 1 1 1  only |Y|=|Z|=V (λ,n,d) permutationsin Sλ are d-close  1 1 1 1 1 1 1 1 1 1  ∞ n   to x.  0 0 1 1 1 1 1 1 1 1    Theorem 1:  0 0 1 1 1 1 1 1 1 1     0 0 0 0 1 1 1 1 1 1  Sλ Sλ   n ≤F (λ,n,d)≤ n .  0 0 0 0 1 1 1 1 1 1  VP∞ro(oλf(cid:12)(cid:12):,nT,(cid:12)(cid:12)do−pro1v)e the∞lower bound,Vw∞e(λu,s(cid:12)(cid:12)en,th⌊(cid:12)(cid:12)ed−2fo1l⌋l)owing Let r(1,m,d) be the row sum of A(1,m,d)’s i-th row. Wehave: i |Sλ| algorithmtogeneratea(λ,n,d)-FPAwithsize≥ n . V∞(λ,n,d−1) d+i if i≤d, 1) C ←∅, D ←Sλ. r(1,m,d) = 2d+1 if d<i≤m−d, n i 2) Add an arbitrary x ∈ D to C, then remove all permu-  m−i+1+d if i>m−d. tations that is (d−1)-close to x from D. Thenfori∈[m]andj ∈[λ],therowsumofthe(iλ−λ+j)-th 3) If D 6=∅ then repeat step 2, otherwise output C. row of A(λ,λm,d) is λr(1,m,d), due to A(λ,λm,d) =A(1,m,d)⊗ D has initially |Sλ| elements and each iteration of step 2 i n 1 . We first calculate V (λ,n,d) by using perA(λ,n,d). removes at most V (λ,n,d − 1), so we conclude |C| ≥ λ ∞ ∞ Lemma 1: |Sλ| n . V∞(λ,n,d−1) perA(λ,n,d) Now we turn to the upper bound. Consider a (λ,n,d)- V (λ,n,d)= . FPAC∗ withthemaximumcardinality.Anytwo⌊d−1⌋-radius ∞ (λ!)n/λ 2 Proof: balls centered at distinct permutations in C∗ do not have any common permutation, since the minimum distance is d. In perA(λ,n,d) other words, the ⌊d−1⌋-radius balls centered at permutations = |{x∈S1 :∀i,a(λ,n,d) =1}| inICt∗isarceleaalrl dthisajtoi|nS2tλ.|W=e haven!|C.∗|It≤isV∞al(rλe,|naSd,nλ⌊y|d−2k1n⌋o).wn that == |({λx!)n∈/λS|nn1{y:m∈aSxnλii,|x:⌈imλi⌉ax−i|⌈⌈xλλii⌉⌉|−≤ydi|}≤| d}| n (λ!)n/λ = (λ!)n/λ|{y ∈Sλ :l (Iλ,y)≤d}| V (1,n,d) equals to the permanent of some special matrix n ∞ n ∞ = (λ!)n/λV (λ,n,d) [8]. In this paper, we generalize previous analysis to give ∞ asymptoticboundsforTheorem1.Thepermanentofann×n The first equality holds since A(λ,n,d) is a (0,1)-matrix and matrix A=(ai,j) is defined as by the definition of permanent. We can convert x ∈ S1 into n n y ∈ Snλ by setting yi = ⌈xλi⌉, and there are exactly (λ!)n/λ perA= ai,πi. x’s in Sn1 converted to the same y. Thus, we know the third πX∈SnYi=1 equalityholds.Therefore,thelemmaholdsbymoving(λ!)n/λ to the left-hand side of the equation. Defineasymmetricn×nmatrixA(λ,n,d) = a(λ,n,d) ,where i,j We still need to estimate perA(λ,n,d) in order to get (cid:16) (cid:17) a(λ,n,d) =1, if ⌈i⌉−⌈j⌉ ≤d; else a(λ,n,d) =0. Note that asymptoticbounds.Kløve [7] reportssome boundsand meth- i,j λ λ i,j a permutation ((cid:12)x ,...,x )(cid:12) is d-close to Iλ if and only if ods to approximate perA(1,n,d). We extend his analysis for (cid:12) 1 n(cid:12) n a(λ,n,d) = 1 for every i ∈ [n]. Now we consider A(λ,λm,d). perA(λ,n,d). Sii,nxcie the λ copies of a symbolare consideredidentical while Lemma 2: perA(λ,n,d) ≤[(2dλ+λ)!]2dλn+λ . computingthedistanceandtheentriesindexedfrom(ℓλ−λ+ Proof: Itisknown(Theorem11.5in[11])thatfor(0,1)- 1)to ℓλof Iλλm representthe same symbolfor everyℓ∈[m]. matrixA,perA≤ ni=1(ri!)r1i whereri isthesumofthei-th Q row.Sincethe sum ofanyrowofA(λ,n,d) is atmost2dλ+λ, Algorithm Enλ,k wehaveperA≤ n [(2dλ+λ)!]2dλ1+λ =[(2dλ+λ)!]2dλn+λ. Input: (m1,...,mk)∈Z2k i=1 Output: (x ,...,x )∈Sλ Q 1 n n WegiveperA(λ,n,d) alowerboundbyusingthevanderWaer- max←n; min←1; for i←1 to k do den permanent theorem (see p.104 in [11]): the permanent of if m =1 an n×n doubly stochastic matrix A (i.e., A has nonnegative i then {x ←⌈max⌉; max←max−1;} entries, and every row sum and column sum of A is 1.) is no i λ mlesastrtihxa,nsinnnnc!e.Uthneforrtouwnasteulmy,sAa(λn,dn,dc)oliusmnontsasduomusblyrasntgoechfarsotmic for i←elske+{x1it←o n⌈mdλoin⌉; min←min+1;} dλ+λ to 2dλ+λ. We estimate the lower boundvia a matrix xi ←⌈mλin⌉; min←min+1; derived from A(λ,n,d) as follows. Output (x1,...,xn). LemPrmoaof:3:LpeteArÃ=(λ,n,d1) ≥A((2λd2,λn2+d,dλλ)),nw·hnincn!h.hasthesumofany Fig.1. Enλ,k encodes messages inZ2k withSnλ. 2dλ+λ row or column bounded by 1, but is not a doubly stochastic matrix. Observe that every row sum of A˜ is 1 except the first Next we investigate the properties of the code obtained by dλ and last dλ rows. For i∈[d] and j ∈[λ], both row (iλ− Eλ . Let Cλ be the image of Eλ . n,k n,k n,k λan+nj×) anndmraotwrix(nB−frioλm+A˜j)wsiuthmetaoch2dd+r+oi1w. Nsuomwewqeuaclotnost1ruacst 2k.Theorem 3: Cnλ,k is a (λ,n,⌊n−λk⌋)-FPA with cardinality follows: Proof: Consider two messages p = (p ,...,p ) and 1 k For i∈[d] and j ∈[λ], add 1 to q =(q1,...,qk)∈Z2k.Letxp andxq betheoutputsofEnλ,k, 2dλ+λ respectively. Let r be the smallest index such that p 6= q . 1) The first (d−i+1)λ entries of row (iλ−λ+j). r r Withoutlossofgenerality,weassumep =1,q =0andthere 2) The last (d−i+1)λ entries of row (n−iλ+j). r r are exactlyz zeroesamongp ,...,p . Consequently,xp is 1 r−1 r The row sums of the first dλ and last dλ rows of B are now set to ⌈max⌉ = ⌈n−r+1+z⌉ and xq is set to ⌈min⌉= ⌈1+z⌉ (d−i+1)λ + d+i =1. by Eλ .λThe distancλe between xrp and xq is: λ λ 2dλ+λ 2d+1 n,k We turn to check the column sums of B. Since A˜ is symmetric and by the definition of B, we know B is sym- n−r+1+z 1+z − metric as well. Thus we have that B is doubly stochastic and (cid:24) λ (cid:25) (cid:24) λ (cid:25) perB ≥ n!. nn n−r+1+z 1+z NowweturntoboundperA(λ,n,d).Observethattheentries > − −1 λ λ of the first dλ andlast dλ rowsof B are atmost 2 times n−r 2dλ+λ = −1 of the corresponding entries of A(λ,n,d), and the other rows λ areexactly 1 timesofthecorrespondingrowsofA(λ,n,d). n−k 2dλ+λ ≥ −1, since r ≤k. We have perA(λ,n,d) ≥ (2dλ+λ)nperB ≥ (2dλ+λ)n n!. λ 22dλ 22dλ nn With Lemma 2 and Lemma 3, we have the asymptotic The first inequality holds by the fact of ceiling function: bounds as follows. a ≤ ⌈a⌉ < a + 1, for any real number a. Note that the Theorem 2: distance has integer value only here. If n−k is integer then λ n! ≤F (λ,n,d)≤ 22λ·⌊d−21⌋nn . wthheicdhistiasncen−iskatelxeaacsttly(cid:4),ni−λ.ek.,(cid:5);theelsedisittainscaetbleeatwstee(cid:6)nn−λakny−t1w(cid:7)o, [(2dλ−λ)!]2dλn−λ ∞ (2λ·⌊d−21⌋+λ)n codewords(cid:4)inλC(cid:5)λ is at least ⌊n−k⌋. Since every message is n,k λ III. ENCODING AND DECODING encoded into a distinct codeword, we have Cnλ,k =2k. Since Cλ is a (λ,n,⌊n−k⌋)-FPA, we let d = ⌊n−k⌋ for Our construction idea is based on the previous work[8] by n,k λ λ convenience. Lin,etal.WegeneralizetheiralgorithmforconstructingFPAs. Furthermore, we give the first locally decoding algorithm for B. Unique decoding algorithm FPAs under l -norm. ∞ Unique decoding algorithms for classic error correcting codesare usuallymuchmorecomplicatedthantheir encoding A. Encoding algorithm algorithms.While,ourproposeddecodingalgorithmUλ (see n,k We give an encoding algorithm Eλ (see Figure 1) which Figure 2) remains simple. n,k convert k-bit message into a permutation in Sλ where n ≥ n k+λ. The running time of Uλ is clearly O(k), even faster than n,k the encoding algorithm. We show its correctness as follows. The encoding algorithm Eλ maps binary vectors from Zk Theorem 4: Given a permutation x = (x ,...,x ) which n,k 2 1 n to Sλ and it is a distance preserving mapping. It is clear that is d−1-close to Eλ (m) for some m ∈ Zk, algorithm Uλ n 2 n,k 2 n,k Eλ runs in O(n) time while encoding any k-bit message. outputs m correctly. n,k Algorithm Unλ,k Proof: By contradiction, assume Lλ does not output Input: (x1,...,xn)∈Snλ before the end of the λ-th iteration. Fonr,kℓ ≤ λ, let jℓ be Output: (m1,...,mk)∈Z2k the index picked in the ℓ-th iteration. For every ℓ ≤ λ, we max←n; min←1; have x = x , otherwise Lλ outputs at the ℓ-th iteration. for i←1 to k do Therefoire, thejℓre are at leastnλ,k+1 entries of x equal to x . i if |txhien−{⌈mmλiax←⌉|1<; m|xaix−←⌈mmλina⌉x| −1;} aItndimLpλliesoxut∈/puStsnλ,ikn,tahecoℓn-tthraditiecrtaiotino.nT. here is some xjℓ 6=xi, n,k else {m ←0; min←min+1;} i Theorem 5: Given a permutationx=(x ,...,x ) δ-close 1 n Output (m1,...,mk). to a codeword Eλ (m) = (y ,...,y ) ∈ Sλ for some m n,k 1 n n,k Fig.2. Unλ,k decodes wordsinSnλ tomessages inZ2k. andanindexi∈[k],Lλn,k outputsmi withprobabilityatleast 1− 2δ+1 at its first iteration. d Proof: Without loss of generality, we assume m = 0, i Proof: By contradiction, assume Uλ outputs mˆ = y =tandletubethemaximumnumberamongy ,...,y , n,k i i+1 n (mˆ ,··· ,mˆ ) 6= m. Let Eλ (m) = (y ,...,y ), r be the i.e., at the start of the i-th iteration min = t and max = u 1 k n,k 1 n smallest index such that m 6= mˆ and z be the number of whileencoding.Assumethereareγ numbersequaltotamong r r zeroes among m1,...,mr−1. At the beginning of the r-th y1,...,yi−1, and there are γ′ numbers equal to u among iteration, max = n−r+1+z and min = 1+z because yi+1,...,yn. According to the encoding algorithm, we have for every i<r, m =mˆ . Without loss of generality, assume 1=m 6=mˆ =0i. Noteithat y is set to ⌈max⌉=⌈n−r+1+z⌉ λ−γ−1 λ γ′ r r r λ λ {y ,...,y }={t,...,t,t+1,...,t+1,...,u,...,u} by Eλ . While mˆ is decoded to 0 by Uλ , we have i+1 n |xr −n⌈,kmλax⌉|≥|xr −r ⌈mλin⌉|. Thus, n,k Since l∞(x,Enλ,k(m)z)≤}|δ{, wze have}||xj −y{j|≤δzan}d||x{i− l∞(x,Enλ,k(m)) ≥ |xr −yr|=|xr −⌈mλax⌉| yi| ≤ δ. The probability that Lλn,k does not output mi at the ≥ 1 |x −⌈max⌉|+|x −⌈min⌉| first iteration is: 2 r λ r λ ≥ 1(cid:0)⌈max⌉−⌈min⌉ (cid:1) Pr[x ≥x ] ≤ Pr[y +δ ≥x ] 2 λ λ i j i j = 1(cid:0)⌈n−r+1+z⌉−⌈1(cid:1)+z⌉ ≥ d. ≤ Pr[y +δ ≥y −δ] 2 λ λ 2 i j (cid:0) (cid:1) = Pr[y +2δ ≥y ]. The last inequality is true, since we know ⌈n−r+1+z⌉ − i j λ ⌈1+z⌉ ≥ n−k = d from the proof of Theorem 3. This There are at most 2δλ+λ−γ−1 possible y ’s less than or λ λ j contradicts(cid:4)that x(cid:5) is d−1-close to Eλ (m). equal to y +2δ. Thus, 2 n,k i C. Locally decoding algorithm (2δ+1)λ−γ−1 2δλ+λ 2δ+1 Pr[x ≥x ]≤ ≤ = . i j NextweshowalocallydecodingalgorithmLλ ,seeFigure n−i dλ d n,k 3,whichis aprobabilisticalgorithm.We discussitsefficiency Therefore, the probability that Lλ outputs m correctly at anderrorprobabilityin thissubsection.We provethatitreads n,k i the first iteration is at least 1− 2δ+1. atmostλ+1entriesof thereceivedwordin Lemma4, hence d Corollary 1: Givenacodewordx=Eλ (m)forsomem its running time is O(λ). It has a chance to output wrongly, n,k and an index i, Lλ outputs m correctly. but we show that the error probability is small in Theorem 5. n,k i Proof: By Lemma 4, there exists ℓ ≤ λ such that Lλ Furthermore,Lλ alwaysoutputscorrectmessage bitwhenit n,k n,k terminates at the ℓ-th iteration. Let j be the index picked at was given a codeword as input, see Corollary 1. the ℓ-th iteration, we have x 6=x , where j >i. Note that x j i is a codeword: x < x implies m = 0 and x > x implies i j i i j Algorithm Lλn,k mi =1. Hence, Lλn,k outputs mi correctly. Input: i∈[n],(x1,...,xn)∈Snλ A private information retrieval system (PIR) consists of Output: mi, the i-th message bit q servers. All servers know a codeword x = (x1,...,xn) J ←{i+1,...,n}; representinga message m=(m ,...,m ), and a user wants 1 k do toknowonebitm ofmviaqueryasymbolfromeachserver. i Uniformly and randomly pick j ∈J; WesayaPIRhasretrievabilityriftheusercanobtainthemes- if xi >xj then output 1; sage bit with probability r. Let D(s,i) be the distribution of if xi <xj then output 0; entryqueriedfromserveri when the user triesto retrievemi. J ←J −{j}; A PIR has privacy p if max ∆(D(s,i),D(s,j)) ≤ i,j∈[k],s∈[q] loop; p, where ∆(·,·) is the statistical distance. A (q,r,p)-PIR is a q-server PIR with retrievability r and privacy p. A (q,r,p)- Fig.3. Lλ decodes onebitbyreadingatmostλ+1symbols. n,k PIR has perfect retrievability if r = 1 and perfect privacy if p=0. Lemma 4: Given a permutation x = (x1,...,xn) ∈ Snλ,k With our FPA Cnλ,k, we construct a (λ+1,1,r)-PIR with and an index i∈[k], Lλ terminates within λ iterations. perfect retrievability and privacy r. The scheme is simple: n,k • For a message m, we put x = Enλ,k(m) on all λ+1 servers. • We retrievemi byLλn,k by queryingentriesfromservers in a random order. The perfect retrievability is guaranteed by Corollary 1. How- ever, in order to retrieve m , x must be queried from some i i servers at certain positions ℓ > i, and we have r > 0. We leave the improvementon the privacy r as our future work. REFERENCES [1] P.Cappelletti,C.Golla,P.Olivo,andE.Zanoni,Flashmemories.Kluwer AcademicPublishers, 1999. [2] J.C. Chang, R. J. Chen, T.Kløve and S. C. Tsai, “Distance-preserving mappingsfrombinaryvectorstopermutations,”IEEETrans.Inform.Th., vol.IT-49,pp.1054–1059,Apr.2003. [3] KlimEfremenko,“3-QueryLocallyDecodableCodesofSubexponential Length,” Electronic Colloquium on Computational Complexity, TR08- 069,2008. [4] SophieHuczynskaandGaryL.Mullen,“Frequencypermutationarrays,” JournalofCombinatorial Designs,vol.14,pp.463–478, 2006. [5] A.Jiang,R.Mateescu,M.SchwartzandJ.Bruck,“RankModulationfor Flash Memories,” in Proc. IEEE Internat. Symp. on Inform. Th., 2008, pp.1731-1735. [6] A. Jiang, M. Schwartz and J. Bruck, “Error-Correcting Codes for Rank Modulation,” in Proc. IEEE Internat. Symp. on Inform. Th., 2008, pp. 1736-1740. [7] Torleiv Kløve, “Spheres of Permutations under the Innity Norm - Per- mutations with limited displacement,” Reports in Informatics, Dept. of Informatics, Univ.Bergen,Reportno.376,2008. [8] T.T.Lin,S.C.TsaiandW.G.Tzeng,“EfficientEncodingandDecoding withPermutationArrays,”inProc.IEEEInternat.Symp.onInform.Th., 2008,pp.211-214. [9] K.W.Shum,”Permutation codingandMFSKmodulation forfrequency selective channel,” IEEEPersonal, Indoor andMobile RadioCommuni- cations, vol.13,pp.2063–2066,Sept.2002. [10] LucaTrevisan,“SomeApplicationsofCodingTheoryinComputational Complexity,” QuadernidiMatematica, vol.13,pp.347-424,2004. [11] J.H. van Lint, R. M. Wilson, A Course in Combinatiorics., 2nd ed. Cambridge, U.K.:CambridgeUniv.Press,2001. [12] A. J. H. Vinck and J. Ha¨ring, “Coding and modulation for power- linecommunications,”inProc.Internat.Symp.onPowerLineCommun., Limerick,Ireland, April2000. [13] A.J.H.Vinck,J.Ha¨ring,T.Wadayama,“CodedM-FSKforpowerline communications,” in Proc. IEEE Internat. Symp. on Inform. Th., 2000, p.137. [14] A. J. H. Vinck, “Coded modulation for powerline communications,” Proc.Int.J.Electron. Commun,vol.54,pp.45-49,2000. [15] Sergey Yekhanin, “Towards 3-query locally decodable codes ofsubex- ponential length,” J.ACM,vol.55(1),pp.1-16,2008.