Polar Codes are Optimal for Write-Efficient Memories

Transcription

1 Polar Codes are Optimal for Write-Efficient Memories Qing Li Department of Computer Science and Engineering Texas A & M University College Station, TX 7784 qingli@cse.tamu.edu Anxiao (Andrew) Jiang Department of Computer Science and Engineering Texas A & M University College Station, TX 7784 ajiang@cse.tamu.edu Abstract Write-efficient memories (WEM) [] were introduced by Ahswede and Zhang as a model for storing and updating information on a rewritable medium with constraints. A coding scheme for WEM using recently proposed polar codes is presented. The coding scheme achieves rewriting capacity, and the rewriting and decoding operation can be done in time O( log ), where is the length of the codeword. I. ITRODUCTIO Write-efficient memories (WEM) are models for storing and updating information on a rewritable medium with constraints. WEM is widely used in data storage area: in flash memories, write-once memories (WOM) [], and the recently proposed compressed rank modulation (CRM) [9] are examples of WEM; codes based on WEM were proposed for phase change memories [8]. The recently proposed scheme, that polar codes are constructed for WOM codes achieving capacity [3], motivates us to construct codes for WEM. A. WEM with a maximal rewriting cost constraint Let X {,,..., q } be the storage alphabet. R + [, + ), and ϕ : X X R + is the rewriting cost function. Suppose that a memory def consists of cells. Given one cell state, x (x, x,..., x ) X, and another cell state, y X, the rewriting cost of changing x to y is ϕ(x, y ) ϕ(x i, y i ). i Let M, and D {,,..., M }. Define the decoding function, D : X D, i.e., D(x ) i. Given the current cell state x, and data to rewrite, j, the rewriting function is defined as R : X D X such that D(R(x, j)) j. Definition. [5] An (, M, q, d) WEM code is a collection of subsets, C {C i : i D}, where C i X, and x C i D(x ) i, such that i j, C i Cj. j, x C, y R(x, j) such that ϕ(x, y ) d. The rewriting rate of an (, M, q, d) WEM code is R log 2 M, and the rewriting capacity function, R(q, d), is the largest d-admissible rate when. Let P(X X ) be the set of joint probability distributions over X X. For a pair of random variables (X, Y ) (X, X ), let P XY denote the joint probability distribution, P Y denotes the marginal distribution, and P Y X denotes the conditional probability distribution. E( ) denotes the expectation operator. If X is uniformly distributed over {,,..., q }, denote it as X U(q). Let P(q, d) {P XY P(X X ) : P X P Y, E(ϕ(X, Y )) d}. R(q, d) is determined as [5]: R(q, d) H(Y X). max P XY P(q,d) For WOM codes, the cell state can only increase but not decrease. WOM codes are special cases of WEM codes: for a WOM cell if we update it from x X to y X, the cost is measured by ϕ(x, y) if y x, and otherwise. Therefore, WOM codes are such WEM codes with ϕ( ) defined previously, and d is equal to. In this paper, we focus on symmetric rewriting capacity function R s (q, d): Definition 2. For X, Y X with P X, P Y and P XY, and ϕ : X X R +, R s (q, d) H(Y X), max P XY P s (q,d) where P s (q, d) def {P XY P(X X ) : P X P Y, X U(q), E(ϕ(X, Y )) d}. We present an example of WEM with R s (q, d) below. (mq)! S q,m is the set of m! permutations over m q m {}}{{}}{ {,,...,,..., q, q,..., q}. We abuse the notation of u qm def [u, u,..., u qm ] to denote an element of S q,m, which denotes the mapping i u i. Example 3. A rewriting code for CRM with a maximal

2 rewriting cost constraint, (q, m, M, d), is defined by replacing X by S q,m, ϕ( ) by Chebyshev distance between u qm, v qm S q,m, d (u qm, v qm ) def max u j v j, and d by d in definition. j {,,...,qm } The above rewriting code is actually an instance of WEM: for x, y X, let ϕ(x, y) if x y d, and otherwise. ow the (q, m, M, d) CRM is an (qm, M, q, ) WEM with X replaced by S q,m, and ϕ( ) is defined previously. B. WEM with an average rewriting cost constraint With deterministic D(x ) and R(x, i), suppose that message sequences M, M 2,..., M t (t ) are written into the memory medium, where M i D is uniformly distributed, represented by x (i), and it forms a markov chain. Its transition probability matrix X is given by µ(y x ) y,x µ(y x ) { M if i s.t.y R(x, i), otherwise. Denote the stationary distribution of x by π(x ). The average rewriting cost D is determined as follows: D lim t t x t i E(ϕ(x (i), x (i + ))), π(x ) ϕ(x, y ), x π(x ) j y µ(y x ) D j (x ), () where D j (x ) is the average rewriting cost of updating from x to data j. Definition 4. An (, M, q, d) ave WEM code is a collection of subsets, C {C i : i D}, where C i X, and x C i D(x ) i, such that i j, C i Cj ; The average rewriting cost D d. The rewriting rate of an (, M, q, d) ave code is R ave log 2 M, and its rewriting capacity function, R ave (q, d), is the largest d-admissible rate when. It is proved that R ave (q, d) R(q, d) []. Similarly, we focus on the symmetric rewriting capacity function, R s ave(q, d), as defined in definition 2. C. The outline The connection between rate-distortion theorem and rewriting capacity theorem is presented in section II; The binary polar WEM codes with an average rewriting cost constraint and a maximal rewriting cost constraint are presented in subsection A and B of section III, respectively; The q-ary polar WEM codes, based on the recently proposed q-ary polar codes [], are presented in subsection A and B of section IV for an average rewriting cost constraint and a maximal rewriting cost constraint, respectively; The conclusion is obtained in section V. II. LOSSY SOURCE CODIG AD ITS DUALITY WITH WEM In this section, we present briefly background of lossy source coding and its duality with WEM, which inspires code constructions for WEM. Let X also denote the variable space, and Y denotes the reconstruction space. Let d : Y X R + denote the distortion function, and the distortion among a vector x and its reconstructed vector y is d(x, y ) i d(x i, y i ). A (q R, ) rate distortion code consists of a encoding function f : X {,,..., q R } and a reproduction function g : {,,..., q R } Y. The associated distortion is defined as E(d(X, g (f (X )))), where the expectation is with respect to the probability distribution on X. R(q, D) is the infimum of rates R such that E(d(X, g (f (X )))) is at most D as. Let P (q, D) def {P XY P(X Y) : E(d(X, Y )) D}, and R(q, D) is min P XY P (q,d) I(X; Y ) [4]. We focus on the double symmetric rate-distortion for (X, Y ) (X X ), and d(x, y), R S (q, D), which is R s (q, D) min I(Y ; X), where P XY P s (q,d) P s (q, D) def {P XY P(X X ) : P X P Y, X U(q), E(d(X, Y )) D}. The duality between R s (q, D) and R s (q, D) is captured by the following lemma, the proof of which is ommited due to being straightforward. Lemma 5. With the same d( ) and ϕ( ), R s (q, D) + R s (q, D) log 2 q. (2) III. POLAR CODES ARE OPTIMAL FOR BIARY WEM Inspired by lemma 5, we show that polar codes can be used to construct binary WEM codes with R s (2, D) in a way related to the code construction for lossy source coding of [7] in this section.

3 A. A code construction for binary WEM with an average rewriting cost constraint ) Background on polar codes [2]: Let W : {, } Y be a binary-input discrete memoryless channel for some output alphabet Y. Let I(W ) [, ] denote the mutual information between the input and output of W with a uniform distribution on the input. Let G denote n-th Kronecker product of ( ). Let the Bhattacharyya parameter Z(W ) WY X (y )W Y X (y ). y X The polar code, C (F, u F ), F {,,..., }, u F denotes the subvector u i : i F, and u F {, } F, is a linear code given by C (F, u F ) {x u G : u F c {, } F c }. The polar code ensemble, C (F ), F {,,..., }, is C (F ) {C (F, u F ), u F {, } F }. The secret of polar codes achieving I(W ) lies in how to select F : define W (i) : {, } Y {, } i as sub-channel i with input u i, output (y, u i W (i) (y, u i u i ) def ) and transition probabilities W (y u ), 2 u i+ where W (y u ) def i W (y i (u G ) i ), and (u G ) i denotes the i-th element of u G ; The fraction of {W (i) } that are approaching noiseless, i.e., Z(W (i) ) 2 β for β 2, approaches I(W ); The F is chozen as indecies with large Z(W (i) ), that is F def {i {,..., } : Z(W (i) ) 2 β } for β /2. The encoding is done by a linear transformation, and the decoding is done by successive cancellation (SC). 2) Polar codes on lossy source coding: We sketch the binary polar code construction for lossy source coding [7] as follows. ote that R s (2, D) can be obtained through the following optimization function: min : I(Y ; X), s.t. : 2 P (y x) x y x y 2 P (x y) 2, P (y x)d(y, x) D. (3) 2 Let P (y x) be the probability distribution minimizing the objective function of (3). P (y x) plays the role of a channel. By convention, we call P (y x) as test channel, and denote it as W (y x). ow, we construct the source code with R s (2, D) using the polar code for W (y x), and denote the rate of the source code by R: set F as ( R) sub-channels with the highest Z(W (i) ), set F c as the remaining R sub-channels, and set u F to an arbitrary value. A source codeword y is mapped to a codeword x C (F, u F ), and x is described by the index u F c (x G ) F c. The reproduction process is done as follows: we do SC encoding scheme, û Û(y, u F ), that is for each k in the range till : ) If k F, set û k u k ; 2) Else, set û k m with the posterior P (m û i, y ). The reproduction codeword is û G. Thus, the average distortion D (F, u F ) (over the source codeword y and the encoder randomness for the code C (F, u F )) is : y P (y ) û F c P (û i û i, y ) i F c d(y, û G ), where û F u F. The expectation of D (F, u F ) over the uniform choice of u F, D (F ), is D (F ) u F 2 F D (F, u F ). Let Q U,Y Q Y (y ) P (y ), and Q U Y Q(u i u i, y ) denote the distribution defined by by { 2, if i F, P (u i u i, y ), otherwise. Thus, D (F ) is equivalent to E Q (d(y, u G )), where E Q ( ) denotes the expectation with respect to the distribution Q U.,Y It is proved that D (F ) D+O(2 β ) for β 2, the rate of the above scheme is R F c, and polar codes achieve the rate-distortion bound by Theorem 3 of [7]. On the other hand, Theorem 2 of [3] further states the strong converse result of the rate distortion theory. More precisely, if y is uniformly distributed over {, }, then δ >, < β < 2, sufficiently large, and with the above SC encoding process and the induced Q, Q(d(u G, y ) D + δ) < 2 β. That is, for y, the above reproduction process obtains x u G such that the distortion d(x, y ) is less than D almost by sure. 3) The code construction: We focus on the code construction with symmetric rewriting cost function, which satisfies x, y, z {, }, ϕ(x, y) ϕ(x + z, y + z), where + is over GF(2).

4 To construct codes for WEM with R s (2, D), we utilize its related form R s (2, D) in (2) and the test channel W (y x) for R s (2, D). ote that W (y x) is a binary symmetric channel. The code construction for (, M, 2, D) ave with rate R is presented in Algorithm III.: Algorithm III. A code construction for (, M, 2, D) ave WEM : Set F as R sub-channels with the highest Z(W (i) ), and set F c as the remaining ( R) sub-channels. 2: The (, M, 2, D) ave code is C {C i : C i C (F, u F (i))}, where u F (i) is the binary representation form of i for i {,,..., M }. Clearly, since G is of full rank [2], u F (i) u F (j), C (F, u F (i)) C (F, u F (j)). The rewriting operation and the decoding operation are defined in Algorithm III.2 and Algorithm III.3, respectively, where the dither g is inspired by [3]. Algorithm III.2 The rewriting operation y R(x, i). : Let v x + g, where g is known both to the decoding function and to the rewriting function, it is chosen such that v is uniformly distributed over {, }, and + is over GF(2). 2: SC encoding v, and this results u Û(v, u F (i)) and ŷ u G. 3: y ŷ + g. Algorithm III.3 The decoding operation u F (i) D(x ). : y x + g. 2: u F (i) (y G ) F. Lemma 6. D(R(y, i)) i holds for each rewriting. Proof: From the rewriting operation, y ŷ + g u G + g Û(v, u F (i))g + g, from the decoding function Û(v, u F (i))g + g + g, which is Û(v, u F (i))g, thus the decoding result is i. 4) The average rewriting cost analysis: From y R(x, i), we know that y Û(v, u F (i))g + g Û(x + g, u F (i))g + g, thus ϕ(x, y ) is ϕ(x, Û(x + g, u F (i))g + g ), which is +g, Û(x +g, u F (i))g ) due to ϕ( ) being symmetric. Denote w x + g, thus ϕ(x, y ) ϕ(w, Û(w, u F (i))g ). The average rewriting cost D lim t t lim t t t i t i Û(w E(ϕ(x (i), x (i + ))), E(ϕ(w,, u F (M i+ ))G )), π(w ) D j (w ). (4) j w Let us focus on D j (w ), which is the average (in this case, over the probability of rewriting to data j and the randomness of the encoder) rewriting cost of updating w to a codeword representing j. Thus D j (w ) P (u 2 F i u i, w )ϕ(w, u G ). u F c i F c Therefore, interpreting ϕ( ) as d( ), D is actually the average distortion over the ensemble C (F ), D (F ). The following lemma from [7] is to bound D: Lemma 7. [7] Let β < 2 be a constant and let σ 2 2 β. When the polar code for the source code with R s (2, D) is constructed with F : F {i {,,..., } : Z(W (i) ) 2σ2 }, then D (F ) D + O(2 β ), where D is the average rewriting cost constraint. Therefore, with the same β, σ, F, and polar code ensemble C (F ), D D + O(2 β ). According to [2], lim lim 2 n,n F c {i {,,..., } : Z(W (i) ) 2 2nβ } I(W ) R s (2, D), thus this implies that for sufficiently large a set F such that F c Rs (2, D) ɛ, ɛ >. In other words, the rate of the constructed WEM code, R F F c Rs (2, D) + ɛ. The complexity of the decoding and the rewriting operation is of the order O( log ) according to [2]. We conclude the theoretical performance of the above polar WEM code as follows:

5 Theorem 8. For a binary symmetric rewriting cost function ϕ : X X R +. Fix a rewriting cost D and < β < 2. For any rate R < Rs (2, D), there exists a sequence of polar WEM codes of length and rate R R, so that under the above rewriting operation, D satisfies D D+O(2 β ). The decoding and rewriting operation complexity of the codes is O( log ). B. A code construction for binary WEM with a maximal rewriting cost constraint The code construction, the rewriting operation and the decoding operation are exactly the same as Algorithm III., Algorithm III.2, and Algorithm III.3, respectively. The rewriting capacity is guaranteed by Lemma 5, the decoding and rewriting operation complexity is the same as polar codes, and the rewriting cost is obtained due to the strong converse result of rate distortion theory, i.e., Theorem 2 of [3]. Thus, we have: Theorem 9. For a binary symmetric rewriting cost function ϕ : X X R +. Fix a rewriting cost D, δ, and < β < 2. For any rate R < Rs (2, D), there exists a sequence of polar WEM codes of length and rate R R, so that under the above rewriting operation and the induced probability distribution Q, the rewriting cost between a current codeword y and its updated codeword x satisfies Q(ϕ(y, x ) D+δ) < 2 β. The decoding and rewriting operation complexity of the codes is O( log ). IV. POLAR CODES ARE OPTIMAL FOR q-ary WEM, q 2 r In this section, we extend the previous binary scheme to q-ary WEM (q 2 r ), considering the length of polar codes, which is 2 n. A. A code construction for q-ary WEM with an average rewriting cost constraint, q 2 r ) Background of q-ary polar codes, q 2 r []: The storage alphabet is X, X q, and for x X, (x, x,..., x r ) is its binary representation. Let W : X Y be a discrete memoryless channel. I(W ) is q W (y x) log W (y x) 2 x X y Y. x q W (y x ) X The sub-channel i, W (i), is defined by W (i) (y, u i u i ) def W (y q u ), where W (y u ) def u i+ W (y i (u G ) i ). i Consider the following bit channel and its Bhattacharyya: Fix k {,,..., r}, and W [r k] (y u) W (y x), where 2 k x:x r k u y Y, u {, } r k, and x (x, x,..., x r ) X ; Define Z(W {x,x }) W (y x)w (y x ), let 2 r x X y Y Z v (W ) Z(W {x,x+v} ) for v X \{}, and Z i (W ) 2 Z i v (W ), where i,,..., r, and v X i X i {v X : i arg max v j } with the binary j r representation of v, (v, v,..., v r ). Z i (W (j) ) i {,,..., r } and j {,,..., } converges to Z i, {, }, and with probability one (Z,, Z,,..., Z r, ) takes one of the values (Z,,..., Z k,, Z k,,..., Z r, ) k,,..., r, i.e., Theorem.b of []. j A k,n iff (Z (W (j) where R k (ɛ) def ), Z (W (j) ),..., Z r (W (j) )) R k(ɛ), ( k D ) ( r D ), D [, ɛ), i ik D ( ɛ, ]. The channel polarizes in the sense that A ki,n i {,,...,} r I(W ) as. For u X, we also represent it by its binary form, that is u (u I(,),..., u I(,r ),..., u I(,),..., u I(,r ) ) {, } r, where u I(i,),..., u I(i,r ) is the binary representation of u i X, and I(i, j) i r + j. i A ki,n, let the frozen bit set be determined by F {I(i, j) : i {,,..., }, j {,,..., k i }} {,,..., r }. Frozen bits for u are defined as u F {, } F with the subvector u i : i F. Finally, the polar code, C (F, u F ), F {,,..., r } and u F {, } F, is a linear code given by C (F, u F ) {x u G : u F c {, } F c }. The polar code ensemble, C (F ), F {,,..., r }, is C (F ) {C (F, u F ) : u F {, } F }. 2) The code construction: We focus on the q-ary WEM code construction with a symmetric rewriting cost function, which satisfies x, y, z {,,..., q }, ϕ(x, y) ϕ(x + z, y + z), where + is over GF(q). Similar to the binary case, to construct codes for WEM with R s (q, D), we utilize its related form R s (q, D) in Lemma 5 and its test channel W (y x). The code construction is presented in Algorithm IV.. Algorithm IV. A code construction for (, M, q, D) ave WEM : i A ki,n, F {I(i, j) : i {,,..., }, j {,,..., k i }}. 2: The (, M, q, D) ave code is C {C i : C i C (F, u F (i))}, where u F (i) is the binary representation form of i for i {,,..., M }.

6 The WEM code rate is R F r, and the polar code rate is R F c r. Similarly, when R approaches R s (q, D), R approaches R s (q, D) based on Lemma 5. The rewriting operation and the decoding operation are defined in Algorithm IV.2 and Algorithm IV.3, respectively, where the SC encoding is a generalization of q-ary lossy source coding, and the dither g is inspired by [3] to make sure the uniform distribution of v. Algorithm R(x, i). IV.2 The rewriting operation y : Let v x + g, where g is known both to the decoding function and to the rewriting function, it is chosen such that v is uniformly distributed, and + is over GF(q). 2: SC encoding v, û Û(v, u F (i)), that is for each k in the range till : û j { u j if j A r,n, m with the posterior P (m û j, v ), where in the above m,,..., q and if j A k,n for k r, the first k bits of û j are fixed, and let ŷ û G. 3: y ŷ g, and is over GF(q). Algorithm IV.3 The decoding operation u F (i) D(x ). : y x + g. 2: u F (i) (y G ) F. The correctness of the above rewriting function can be verified similarly to Lemma 6. 3) The average rewriting cost analysis: Similar to the analysis of equation (4), we obtain that D π(w ) D j (w ). j w In the following, we first focus on D j (w ). ote that û Û(v, u F (j)) is random, i.e., the SC encoding function may result in different outputs for the same input. More precisely, in step i of the SC encoding process, i r A k,n, û i m with the posterior k P (m û i, v ), where if i A k,n, the first k bits of û i are fixed and known. This implies that the probability of picking a vector u with Âr,n given v with A r,n is equal to if  r,n A r,n, i r P (u i u i, v ) otherwise, A k,n k where in the second case i A k,n, the first k bits of u i are fixed. Therefore, the average (in this case, over the probability of rewriting to data j and the randomness of the encoder) rewriting cost of updating w to a codeword representing j, Dj (w 2 F u F c ), is i r k A k,n ϕ(w, u G ), w P (u i u i, w ) where u F u F (j), u F c {, } F c, and the summation over u F c takes care of A r,n and the fact that i A k,n, the first k bits of u i are fixed. Thus, we obtain that D π(w ) D j (w ), j w π(w ) u F (j) P (u i u i, w < π(w w 2 F )ϕ(w ) q Ar,n u c F i r k A k,n, u G ), u i r k A k,n P (u i u i, w )ϕ(w, u G ). (5) Let Q U,W Q W by denote the distribution defined by defined (w ) π(w ), and Q U W Q(u i u i, w ) { q if i A r,n, P (u i u i, w ) otherwise. Then, inequation (5) is equivalent to D < E Q (ϕ(w, u G )), where E Q ( ) denotes the expectation with respect to the distribution Q U,W. Similarly, let E P ( ) denote the expectation with respect to the distribution P U,W. The following three lemmas are already proved in [7] for q 2 and in [6] for primary q, they extend trivially to q 2 r, and we omit their proofs. Lemma. Q(w, u ) P (w, u ) P (u i u i, w ) ). w,u q i A r,n u i E P ( q

7 Lemma. Let F be chosen such that for i A r,n, q u i E P ( q P (u i u i, w ) ) σ. Then, the average rewriting cost is bounded by E Q(ϕ(w, u G )) E P (ϕ(w, u G )) + A r,n d max σ, where d max def max x,y ϕ(x, y). Lemma 2. E P (ϕ(w, u G )) D. The following lemma, which is a modification of Lemma 5 [6], presents that it is sufficient to choose the set F as those bits with indexes I(i, j) for which i {,,..., }, j {,,..., k i } with i A ki,n. Lemma 3. If i A r,n, that is (Z (W (i) ),..., Z r (W (i) )) R r(ɛ), and let ɛ r ɛ, then q u i E P ( q P (u i u i, w ) ) 2ɛ. Proof: By Pinsker s inequality, for two distribution functions P and Q defined on X, P Q 2D(P Q), where D(P Q) is the Kullback-Leibler divergence between two distributions, that is D(P Q) i log 2( P (i) )P (i), we obtain that: q u i Q(i) E P q P (u i u i, w ) P (u i) q w,u i P (u i )P (u i, w ) P (u i u i, w )P (u i, w ), P (u i )P (u i, w ) w,u i P (u i, w ), 2 log 2 w,u i P (u i, w ), 2I(W (i) ), 2ɛ, P (u i, w ) P (u i )P (u i, w ) where the last inequality is based on the conclusion of Lemma [], that is if (Z (W (i) ),..., Z r (W (i) )) R k (ɛ) k,,..., r, then I(W (i) ) (r k) γ with γ max(k ɛ, (2 r k )ɛ log 2 e). Therefore, the performance of the above polar WEM code can be concluded as follows: Theorem 4. For a q-ary (q 2 r ) symmetric rewriting cost function ϕ : X X R +. Fix a rewriting cost D and < β < 2. For any rate R < Rs (q, D), there exists a sequence of polar WEM codes of length and rate R R, so that under the above rewriting operation, D satisfies D D+O(2 β ). The decoding and rewriting operation complexity of the codes is O( log ). B. A code construction for q-ary WEM with a maximal rewriting cost constraint, q 2 r Similarly, the code construction, the rewriting operation and the decoding operation are exactly the same as Algorithm IV., Algorithm IV.2, and Algorithm IV.3, respectively. ext, we mainly focus on its performance. Theorem 5. For a q-ary (q 2 r ) symmetric rewriting cost function ϕ : X X R +. Fix a rewriting cost D, δ, and < β < 2. For any rate R < Rs (q, D), there exists a sequence of polar WEM codes of length and rate R R, so that under the above rewriting operation and the induced probability distribution Q, the rewriting cost between a current codeword y and its updated codeword x satisfies Q(ϕ(y, x ) D+δ) < 2 β. The decoding and rewriting operation complexity of the codes is O( log ). Proof: We mainly focus on the rewriting cost analysis. The proof of this part is based on the ɛ-strong typical sequence [4] and Theorem 4 and 5 of [3]. We give the sketch as follows. We recall ɛ-strong typical sequences x y X Y with respect to p(x, y) over X Y, and denote it by A () ɛ (X, Y ). We denote by C(a, b x, y ) the number of occurrences of a, b in x, y with the same indexes, and require the following. First, a, b X Y with p(a, b) >, C(a, b x, y )/ p(a, b) < ɛ. Second, a, b X Y with p(a, b), C(a, b x, y ). In our case y u G. Due to the full rank of G, there is a one-to-one correspondence between u and y. We say that u, x A () ɛ (U, X) if x, u G Aɛ () (X, Y ) with respect to q W (y x), where W (y x) is the test channel. The first conclusion is that for sufficiently large, Q(A () ɛ (U, X)) > 2 β for < β < 2, ɛ >, which is a generalization of Theorem 4 of [3], and where Q(A () ɛ (U, X)) Q( a, b :

8 C(a, b u G, x ) q W (a b) ɛ). The outline is that based on Lemma and Lemma we obtain that u,x Aɛ () (U,X) Q(u, x ) P (u, x ) A r,n σ d max. Thus, we obtain Q(u, x ) P (u, x ) u,x u,x A r,n σ d max, Q(u, x ) P (u, x ), where in the above u, x A () ɛ (U, X). By lower bounding P (A () ɛ (U, X)) P ( a, b : C(a, b u G, x ) q W (a b) ɛ) 2q 2 e 2ɛ2 based on Hoeffding s inequality, and (U, X)) P (A () ɛ (U, X)) A r,n σ d max, Q(A () ɛ we obtain the desired result by setting σ 2 β The second conclusion is that let y R(x, i), then δ >, < β < 2 and sufficiently large, Q(ϕ(y, x )/ D + δ) < 2 β, which is a generalization of Theorem 5 of [3]. The outline is that 2d max. Q(ϕ(x, y )/ D + δ) Q(ϕ(x, y )/ D + δ x, y A () ɛ (X, Y )) + Q(x, y A () ɛ (X, Y )), 2 β, where the last inequality is based on the conclusion just obtained Q(x, y A () ɛ (X, Y )) < 2 β, and that when x, y Aɛ () (X, Y ), for ɛ sufficiently small and sufficiently large, ϕ(y, x )/ D + δ. V. COCLUSIO Code constructions for WEM using recently proposed polar codes have been presented. Future work focuses on exploring error-correcting codes for WEM. VI. ACKOWLEDGEMET This work was supported in part by the ational Science Foundation (SF) CAREER Award CCF and the SF Grant CCF REFERECES [] R. Ahlswede and Z. Zhang, Coding for write-efficient memory, Inform. Comput, vol. 83, no., pp. 8 97, Oct 989. [2] E. Arikan, Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels, IEEE Trans on Inf Theory, vol. 55, no. 7, pp , July 29. [3] D. Burshtein and A. Strugatski, Polar write once memory codes, in Proceedings of the 22 IEEE international conference on Symposium on Information Theory (ISIT 2), Boston, MA, July 22, pp [4] T. M. Cover and J. A. Thomas, Elements of Information Theory. ew York: Wiley, 99. [5] F. Fu and R. W. Yeung, On the capacity and error-correcting codes of write-efficient memories, IEEE Trans on Inf Theory, vol. 46, no. 7, pp , ov 2. [6] M. Karzand and E. Teletar, Polar codes for q-ary source coding, in Proc. IEEE International Symposium on Information Theory (ISIT), Austin, Texas, June 2, pp [7] S. B. Korada and R. Urbanke, Polar codes are optimal for lossy source coding, IEEE Trans. Inf. Theory, vol. 56, no. 4, pp , April 2. [8] L. A. Lastras-Montano, M. Franceschini, T. Mittelholzer, J. Karidis, and M. Wegman, On the lifetime of multilevel memories, in Proceedings of the 29 IEEE international conference on Symposium on Information Theory (ISIT 9), Coex, Seoul, Korea, 29, pp [9] Q. Li, Compressed rank modulation, in Proc. 5th Annual Allerton Conference on Communication, Control and Computing (Allerton), Monticello, IL, October 22. [] W. Park and A. Barg, Polar codes for q-ary channels, q 2 r, in Proc. IEEE International Symposium on Information Theory (ISIT), Boston, MA, June 22, pp [] R. L. Rivest and A. Shamir, How to reuse a write-once memory, Inf. Contr, vol. 55, pp. 9, 982.