2011 IEEE Iteratioal Symposium o Iformatio Theory Proceedigs Nouiform Codes for Correctig Asymmetric Errors Hogchao Zhou Electrical Egieerig Departmet Califoria Istitute of Techology Pasadea, CA 91125 Email: hzhou@caltechedu Axiao (Adrew Jiag Computer Sciece ad Egieerig Departmet Texas A&M Uiversity College Statio, TX 77843 Email: aiag@csetamuedu Jehoshua Bruck Electrical Egieerig Departmet Califoria Istitute of Techology Pasadea, CA 91125 Email: bruck@caltechedu Abstract Codes that correct asymmetric errors have importat applicatios i storage systems, icludig optical disks ad Read Oly Memories The costructio of asymmetric error correctig codes is a topic that was studied extesively, however, the existig approach for code costructio assumes that every codeword could sustai t asymmetric errors Our mai observatio is that i cotrast to symmetric errors, the error probability of a codeword is cotext idepedet (sice the error probability for 1s ad 0s is idetical, asymmetric errors are cotext depedet For example, the all-1 codeword has a higher error probability tha the all-0 codeword (sice the oly errors are 1 0 We call the existig codes uiform codes while we focus o the otio of ouiform codes, amely, codes whose codewords ca tolerate differet umbers of asymmetric errors depedig o their Hammig weights The goal of ouiform codes is to guaratee the reliability of every codeword, which is importat i data storage to retrieve whatever oe wrote i We prove a almost explicit upper boud o the size of ouiform asymmetric error correctig codes ad preset two geeral costructios We also study the rate of ouiform codes compared to uiform codes ad show that there is a potetial performace gai I INTRODUCTION Asymmetric error-correctig codes have importat applicatios i storage ad commuicatio systems, such as optical fibers, optical disks, VLSI circuits ad Read Oly Memories I such systems, the error probability from 1 to 0 is sigificatly higher tha the error probability from 0 to 1, which is modeled by biary asymmetric chael (the Z chael the trasmitted sequeces oly suffer oe type of errors, say 1 0 Asymmetric error-correctig codes have bee widely studied: I [1], Kløve summarized ad preseted several such codes I additio, a large amout of effort is cotributed to the desig of systematic codes [2], [3], costructig sigle or multiple error-correctig codes [4] [6], icreasig the lower bouds [7] [9] ad applyig LDPC codes i the cotext of asymmetric chaels [10] However, the existig approach for code costructio is similar to the approach take i the costructio of symmetric error correctig codes, amely, it assumes that every codeword could sustai t asymmetric errors As a result, differet codewords might have differet reliability To see this, let s cosider errors to be iid, every bit that is a 1 ca chage to a 0 by a asymmetric error with crossover probability p > 0 For a codeword x = (x 1, x 2,, x {0, 1}, let w(x = {i : 1 i, x i = 1} deote the Hammig weight of x The the probability for x to have at most t asymmetric errors is P t (x = P (t, w(x, p, P (t, m, p t i=0 ( m p i (1 p m i i Sice x ca correct t errors, P t (x is the probability of correctly decodig x (assumig codewords with more tha t errors are ucorrectable It ca be readily observed that the reliability of codewords decreases whe their Hammig weights icrease Differet from telecommuicatio applicatios, i data s- torage we care about the worst-case performace, amely, we eed guaratee that every codeword ca be correctly decoded with very high probability I this case, it is ot desired to let all the codewords tolerate the same umber of asymmetric errors, sice the codeword with the highest Hammig weight will become a bottleeck ad limit the code rate This motivated us to propose the cocept of ouiform codes, whose codewords ca tolerate differet umbers of asymmetric errors based o their Hammig weights The obective is to guaratee the reliability of every codeword That is, we cosider the worst-case istead of the average-case reliability of the codewords Give this costrait, we would like to maximize the size of the code Specifically, let q e < 1 to be maximal tolerated error probability for each codeword ad let t(x deote the umber of asymmetric errors that x ca correct The give a code C, for every codeword x C, we have P (t(x, w(x, p 1 q e, so that every erroeous codeword ca be corrected with probability at least 1 q e The rest of the paper is orgaized as follows I Sectio II, we provide some defiitios ad properties related to ouiform codes I Sectio III, we give a almost explicit upper boud for the size of ouiform codes Two geeral costructios, based o multiple layers or bit flips, are proposed i Sectio IV ad Sectio V Fially, Sectio VI studies the asymptotic rates of ouiform codes ad uiform codes (both upper bouds ad lower bouds A exteded versio of this paper with detailed proofs ad explaatios is give i [16] 978-1-4577-0594-6/11/$2600 2011 IEEE 1011
II DEFINITIONS AND PROPERTIES A code C is called a ouiform (, p, q e code if for each codeword x C, it ca correct t(w(x asymmetric errors, t(w = mi{s N P (s, w, p 1 q e } (1 That implies each codeword i C ca be recovered with probability at least 1 q e The maximum size of a ouiform (, p, q e code is deoted by B β (, p, q e As compariso, most existig error-correctig codes are uiform codes For a code C of codeword legth, the Hammig weight of its codewords is at most (Ad i may existig asymmetric error-correctig codes, the maximum codeword weight ideed equals [1] So we defie C to be a uiform (, p, q e code if every codeword ca correct t asymmetric errors, t = t( = mi{s N P (s,, p 1 q e } The maximum size of a uiform (, p, q e code is deoted by B α (, p, q e Lemma 1 For ay 0 < p, q e < 1 ad iteger w i [0, ], we have 0 t(w + 1 t(w 1 for a ouiform (, p, q e code Give two biary vectors x = (x 1,, x ad y = (y 1,, y, we say x y if ad oly if x i y i for all 1 i Let S s (x be the set of vectors obtaied by chagig at most s 1 s i x ito 0 s, ie, S s (x = {v {0, 1} v x ad N(x, v s} N(x, y {i : x i = 1, y i = 0} Let S s,s(x be the set of vectors obtaied by chagig at most s 0 s i x ito 1 s or at most s 1 s i x ito 0 s, ie, S s,s(x = {v {0, 1} v x ad N(x, v s} {v {0, 1} x v ad N(v, x s } Note that S s (x = S 0,s (x The followig properties of ouiform codes ca be easily proved, as the geeralizatios of those for uiform codes, icludig Lemmas 22, 23, 32, 33 i [1] Lemma 2 Code C is a ouiform (, p, q e code if ad oly if S t(w(x (x S t(w(y (y = ø for all x, y C with x y Lemma 3 There always exists a ouiform (, p, q e code of the maximum size that cotais the all-zero codeword Give a ouiform code C, let C r deote the umber of codewords with Hammig weight r i C, ie C r = {x C w(x = r} Lemma 4 Let C be a ouiform (, p, q e code ad t(w is defied i (1 For iteger r i [0, ], let s be a iteger such that 0 s t(r s ad let k = max{z 0 z, z (t(z s r}, the we have s ( r + =1 C r + t(k s ( r + C r+ ( r Note that i Lemma 4, if we let s = 0, the we ca get t(k ( r + C r+ ( r k = max{z 0 z, z t(z r} This iequality will be used to get a almost explicit upper boud for the size of ouiform codes III AN ALMOST EXPLICIT UPPER BOUND We ow derive a almost explicit upper boud for the size of ouiform codes, followed the idea of Kløve [11] for uiform codes First, we defie h(r = max{w 0 w, w t(w = r}, h(r = mi{w 0 w, w t(w = r} Ad let M β (, p, q e = max z r, the maximum is take over the followig costraits: 1 z r are o-egative real umbers; 2 z 0 = 1; 3 t(h(r ( r+ zr+ ( r for r 0 The M β (, p, q e is a upper boud for B β (, p, q e Here, coditio 2 is give by Lemma 3, ad coditio 3 is give by Equ (2 from Lemma 4 Our goal i this sectio is to fid a almost explicit way to express M β (, p, q e Lemma 5 Assume z r is maximized over z 0, z 1,, z i the problem above Let Z r = mi{ r,t(h(r} The Z r = ( r for r t( ( r + z r+ Proof: Suppose that Z r < ( r for some r t( Let g = h(r ad k = mi{w z w > 0, w > g} Let m = max{w k t(k > w} The it ca be proved that for all r < w m, Z w < ( w Now, we costruct a ew group of real umbers z0, z1,, z such that 1 zg = z g + 2 zk = z k δ 3 zr = z r for r h, r k with w Zw = mi({( ( g r w m} w k w {( ( g k m < w g}, wz 1 δ = mi{ ( w k ( w m < w g} g (2 1012
For such, δ, it is ot hard to prove that Zr 0 r O the other had, zr = z r + δ > z r, = ( r for which cotradicts our assumptio that z r is maximized over the costrais So the lemma is true Similarly, usig the same idea as above, we ca get the followig lemma Lemma 6 Assume z r is maximized over z 0, z 1,, z i the problem above Let Y r = mi{ r,t(h(r} The Y r = ( r for r t( ( r + z r+ Now let y 0, y 1,, y be a group of optimal solutios to z 0, z 1,, z that maximize z r The y 0, y 1,, y satisfy the coditio i Lemma 6 We see that y 0 = 1 The based o Lemma 6, we ca get y 1,, y uiquely by iteratio Hece, we have the followig theorem for the upper boud M β (, p, q e Theorem 7 Let y 0, y 1,, y be defied by 1 y 0 = 1; 2 y r = 0, 1 r max{s 1 s, s t(s}; 3 y r = ( t(r 1 [( t(r r r t(r =1 y ( r r t(r ], max{s 1 s, s t(s} < r The B β (, p, q e M β (, p, q e = y r This theorem provides a almost explicit expressio for the upper boud M β (, p, q e, which is much easier to calculate tha the equivalet expressio defied at the begiig of this sectio IV CONSTRUCTIONS BASED ON MULTIPLE LAYERS I [1], Kløve summarized some costructios of uiform codes for correctig asymmetric errors The code of Kim ad Freima was the first code costructed for correctig multiple asymmetric errors Varshamov [12] ad Costrai ad Rao [13] preseted some costructios based group theory Later, Delsarte ad Piret [14] proposed a costructio based o expurgatig/pucturig with some improvemets give by Weber et al [15] I this sectio, we propose a geeral costructio of ouiform codes based o multiple layers From the defiitio of ouiform codes, we kow that t(w ca be easily ad uiquely determied by p, q e So a questio arises: if, t(w (for 0 w are give, how to costruct a ouiform code efficietly? Ituitively, we ca divide all the codewords of a ouiform code ito at most t( + 1 layers such that all the codewords i the i th layer (with 0 i t( ca tolerate at least i asymmetric errors I other words, the code is the combiatio of up to t( + 1 uiform codes, each of which corrects a differet umber of asymmetric errors However, we caot desig such a code by costructig codewords idepedetly for differet layers, because a simple combiatio of several idepedet codes may violate the error correctio requiremets of the ouiform codes, due to the iterferece betwee two eighbor layers Our idea is simple: let s first costruct a code which ca tolerate t( asymmetric errors The we add some codewords to the lowest t( layers such that the codewords i the top layer keep uchaged ad they still ca tolerate t( asymmetric errors, ad the codewords i the other layers ca tolerate up to t( 1 asymmetric errors Iteratively, we ca cotiue to add may codeword ito the lowest t( 1 layers Based o this idea, give, t(w, we costruct layered codes as follows Theorem 8 (Layered Codes Let k = t( ad let C 0, C 1,, C k be k + 1 biary codes of codeword legth, C 0 C 1 C k ad for 0 t k, the code C t ca correct t asymmetric errors Let C = {x {0, 1} x C t (w(x}, t (w(x = t(max{w w t(w w(x} The for all x C, x ca tolerate t(w(x asymmetric errors Proof: We prove that for all x, y C with x y, S t(w(x (x S t(w(y (y = ø Wlog, we assume w(x w(y If w(x t(w(x > w(y, the coclusio is true If w(x t(w(x w(y ad w(x w(y, we have S t(w(x (x S t(w(y (y S t (w(y(x S t (w(y(y However, we kow that x C t (w(x C t (w(y ad y C t (w(y, therefore S t (w(y(x S t (w(y(y = ø Furthermore, we have S t(w(x (x S t(w(y (y = ø We see that the costructios of layered codes are based o the provided group of codes C 0, C 1,, C k such that C 0 C 1 C k ad for 0 t k, the code C t ca correct t asymmetric errors Examples of such codes iclude Varshamov codes [12], BCH codes, etc Oe costructios of BCH codes ca be described as follows: Let (α 0, α 1,, α 1 be distict ozero elemets of G 2 m with = 2 m 1 For 0 t k, let C t := {x {0, 1} x i α (2l 1 i = 0 for 1 l t} i=1 I the above example, assume x is a codeword i C t ad y = x + e is a received word with error e, the there is a efficiet algorithm to decode y ito a codeword, which is deoted by D t (y If y has at most t asymmetric errors, the D t (y = x I the followig theorem, we show that the layered codes proposed above also have a efficiet decodig algorithm if D t ( (for 0 t k are provided ad efficiet Theorem 9 (Decodig of Layered Codes Let C be a layered code, let x C be a codeword, ad let y = x + e be a received word such that e = N(x, y t(w(x (Here e is the asymmetric-error vector The there exists at least oe iteger t such that 1013
1 t (w(y t t (w(y + t (w(y; 2 D t (y C; 3 y D t (y ad N(D t (y, y t(w(d t (y For such t, we have D t (y = x Proof: If we let t = t (w(x, the we ca get that t satisfies the coditios ad D t (y = x So such t exists Now we oly eed to prove that oce there exists t satisfyig the coditios i the theorem, we have D t (y = x We prove this by cotradictio Assume there exists t satisfyig the coditios but z = D t (y x The N(z, y t(w(z ad N(x, y t(w(x, which cotradicts the property of the layered codes Accordig to the above theorem, to decode a oisy word y, we ca check all the itegers betwee t (w(y ad t (w(y+ t (w(y to fid the value of t Oce we fid the iteger t satisfyig the coditios i the theorem, we ca decode y ito D t (y directly (Note that t (w(y + t (w(y t (w(y is ormally much smaller tha w(y It is approximately p 2 (1 p 2 w(y whe w(y is large We see that this decodig process is efficiet if D t ( is efficiet for 0 t k V CONSTRUCTIONS BASED ON BIT FLIPS May o-liear codes desiged to correct asymmetric errors do ot yet have efficiet ecodig algorithms Namely, it is ot easy to fid a efficiet ecodig fuctio f : {0, 1} k C with k log C O the other had, i [12], Varshamov showed that liear codes have early the same ability to correct asymmetric errors ad symmetric errors (for the uiform code case I this subsectio, we focus o the approach of desigig ouiform codes for asymmetric errors with efficiet ecodig schemes, by utilizig the well studied liear codes for symmetric errors We ca use a liear code to correct t( asymmetric errors directly, but this method is iefficiet ot oly because the decodig sphere for symmetric errors is greater tha the sphere for asymmetric errors (ad therefore a overkill, but also because for low-weight codewords, the umber of asymmetric errors they eed to correct ca be much smaller tha t( Our idea is to build a flippig code that uses oly low-weight codewords (specifically, codewords of Hammig weight o more tha 2, because they eed to correct fewer asymmetric errors ad therefore ca icrease the code s rate I the rest of this sectio, we preset two differet costructios A First Costructio First, costruct a liear code C (like BCH codes of legth with geerator matrix G that corrects t( 2 symmetric errors Assume the dimesio of the code is k For ay biary message u {0, 1} k, we ca map it to a codeword x i C such that x = ug Next, let x deote a word obtaied by flippig all the bits i x such that if x i = 0 the x i = 1 ad if x i = 1 the x i = 0; ad let y deote the fial codeword correspodig to u We check whether w(x > 2 ad costruct y i the followig way: { x000 if w(x > y = 2 x111 otherwise Here, the auxiliary bits (0 s or 1 s are added to distiguish that whether x has bee flipped or ot, ad they form a repetitio code to tolerate errors The correspodig decodig process is straightforward: Assume we received a word y If there is at least oe 1 i the auxiliary bits, the we flip the word by chagig all 0 s to 1 s ad all 1 s to 0 s; otherwise, we keep the word uchaged The we apply the decodig scheme of the code C to the first bits of the word Fially, the message u ca be successfully decoded if y has at most t( 2 errors i the first bits B Secod Costructio I the previous costructio, several auxiliary bits are eeded to protect oe bit of iformatio, which is ot very efficiet I this sectio, we try to move this bit ito the message part of the codewords i C This motivates us to give the followig costructio Let C be a liear code with legth that corrects t symmetric errors (we will specify t later Assume the dimesio of the code is k Now, for ay biary message u {0, 1} k 1 of legth k 1, we get u = 0u by addig oe bit 0 i frot of u The we ca map u to a codeword x i C such that x = (0uG = 0uv G is the geerator matrix of C i systematic form ad the legth of v is k Let α be a codeword i C such that the first bit α 1 = 1 ad its weight is the maximal oe amog all the codeword i C, ie, α = arg max w(x x C,x 1 =1 Geerally, w(α is very close to I order to reduce the weights of the codewords, we use the followig operatios: Calculate the relative weight w(x α = {1 i x i = 1, α i = 1} The we get the fial codeword { x + α if w(x α > w(α y = 2 x otherwise + is the biary sum, so x + α is to flip the bits i x correspodig the oes i α So far, we see that the maximal weight for y is w(α 2 That meas we eed to select t such that t = t( w(α 2 I the above ecodig process, for differet biary messages, they have differet codewords Ad for ay codeword y, we have y C That is because either y = x or y = x + α, both x ad α are codewords i C ad C is a liear code The decodig process is very simple: Give the received word y = y + e, we ca always get y by applyig the decodig scheme if e t If y 1 = 1, that meas x has bee flipped based o α, so we have x = y + α; otherwise, x = y The the iitial message u = x 2 x 3 x k 1014
Lower Boud Upper Boud η α (, p, q e [1 H(2p]I 0 p 1 4 (1 + p[1 H( p 1+p ] η β (, p, q e max 0 θ 1 p H(θ θh(p (1 θh( pθ 1 θ TABLE I max 0 θ 1 H((1 pθ θh(p C Commets Whe is sufficietly large, the codes based o flips above become early as efficiet as a liear codes correctig t( 2 symmetric errors (We defie the codes efficiecy i Sectio VI It is much more efficiet tha desigig a liear code correctig t( symmetric errors Note that whe is large ad p is small, these codes ca have very good performace o efficiecy That is because whe is sufficietly large, the efficiecy of a optimal ouiform code is domiated by the codewords with the same Hammig weight w d ( 2, ad w d approaches 2 as p gets close to 0 We ca ituitively uderstad it based o two facts whe is sufficietly large: (1 There are at most 2 (H( w d +δ codewords i this optimal ouiform code (2 Whe p becomes small, we ca get a ouiform code with at least 2 (1 δ codewords So whe is sufficietly large ad p is small, we have w d 2 Hece, the optimal ouiform code has almost the same asymptotic efficiecy with a optimal weight-bouded code (Hammig weight is at most /2, which corrects t(/2 errors Beside simplicity ad efficiecy, aother advatage of these codes is that they do ot require the Z-chael to be perfect, ie, it is allowed to have 0 1 errors with very small probability (as log as this probability is smaller tha the probability of 1 0 errors All these properties make these codes very useful i practice However, whe p is ot small, how to desig efficiet ouiform codes with simple ecodig/decodig schemes is still a ope problem VI BOUNDS ON THE RATE Give (, p, q e, we ca defie the efficiecy of uiform codes as η α (, p, q e log 2 B α(,p,q e ad defie the efficiecy of ouiform codes as η β (, p, q e log 2 B β(,p,q e I this sectio, give 0 < p, q e < 1, we study the asymptotic behavior of η α (, p, q e ad η β (, p, q e as Table I summarizes the upper bouds ad lower bouds of η α (, p, q e ad η β (, p, q e obtaied i our full paper [16] We plot them i Fig 1 The gap betwee the bouds for the two codes idicates the potetial improvemet i efficiecy by usig the ouiform codes (compared to usig uiform codes whe the codeword legth is large ACKNOWLEDGMENT This work was supported i part by the NSF CAREER Award CCF-0747415, the NSF grat ECCS-0802107, ad by a NSF-NRI award REFERENCES [1] T Kløve, Error correctig codes for the asymmetric chael, Techical Report, Dept of Iformatics, Uiversity of Berge, 1981 (Updated i 1995 Efficiecy η 1 09 08 07 06 05 04 03 02 01 Bouds for uiform codes Bouds for ouiform codes 0 10 4 10 3 10 2 10 1 10 0 Crossover probability p Fig 1 Bouds to η α (, p, q e ad η β (, p, q e The dashed curves represet the lower ad upper bouds to η α(, p, q e, ad the solid curves represet the lower ad upper bouds to η β (, p, q e [2] K A S Abdel-Ghaffar ad H C Ferreira, Systematic ecodig of the Varshamov-Teegol ts codes ad the Costati-Rao codes, IEEE Tras Iform Theory, vol 44, pp 340-345, Ja 1998 [3] B Bose ad S Al-Bassam, O systematic sigle asymmetric errorcorrectig codes, IEEE Tras Iform Theory, vol 46, pp 669-672, Mar 2000 [4] S Al-Bassam, R Vekatesa, ad S Al-Muhammadi, New sigle asymmetric error-correctig codes, IEEE Tras Iform Theory, vol 43, pp 1619-1623, Sept 1997 [5] Y Saitoh, K Yamaguchi, ad H Imai, Some ew biary codes correctig asymmetric/uidirectioal errors, IEEE Tras Iform Theory, vol 36, pp 645-647, May 1990 [6] L G Tallii, B Bose, O a ew class of error cotrol codes ad symmetric fuctios, IEEE Iteratioal Symposium o Iformatio Theory (ISIT, pp 980-984, 2008 [7] T Etzio, Lower bouds for asymmetric ad uidirectioal codes, IEEE Tras Iform Theory, vol 37, pp 1696-1704, Nov 1991 [8] Z Zhag ad X Xia, New lower bouds for biary codes of asymmetric distace two, IEEE Tras Iform Theory, vol 38, pp 1592-1597, Sept 1992 [9] F Fu, S Lig, ad C Xig, New lower bouds ad costructios for biary codes correctig asymmetric errors, IEEE Tras Iform Theory, vol 49, pp 3294-3299, Dec 2003 [10] C Wag, S R Kulkari, ad H V Poor, Desity evolutio for asymmetric memoryless chaels, IEEE Tras Iform Theory, vol 51, pp 4216-4236, Dec 2005 [11] T Kløve, Upper bouds o codes correctig asymmetric errors, IEEE Tras Iform Theory, vol 27, o 1, pp 128-131, 1981 [12] R R Varshamov, A class of codes for asymmetric chaels ad a problem from the additive theory of umbers, IEEE Tras Iform Theory, vol 19, o 1, pp 92-95, 1973 [13] S D Costati ad T R N Rao, O the theory of biary asymmetric error-correctig codes, Iform Cotr, vol 40, pp 20-36, 1979 [14] P Delsarte ad P Piret, Bouds ad costructios for biary asymmetric error-correctig codes, IEEE Tras Iform Theory, vol 27, pp 125-128, 1981 [15] J H Weber, C de Vroedt, ad D E Boekee, Bouds ad costructios for biary codes of legth less tha 24 ad asymmetric distace less tha 6, IEEE Tras Iform Theory, vol 34, pp 1321-1331, Sept 1988 [16] H Zhou, A Jiag, ad J Bruck, Nouiform codes for correctig asymmetric errors i data storage, Techical Report, Califoria Istitute of Techology, http://paradisecaltechedu/etrhtml, 2011 1015