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1 Reduced-Redundancy Poduct Codes fo Bust Eo Coection Ron M. Roth Gadiel Seoussi y Abstact Inatypical bust-eo coection application of a poduct code of n v n h aays, one uses an [n h ;n h h ]codec h that detects coupted ows, and an [n v ;n v v ]codec v that is applied to the columns while egading the detected coupted ows as easues. Although this conventional poduct code scheme oes vey good eo potection, it contains excessive edundancy, due to the fact that the code C h povides the code C v with infomation on many eo pattens that exceed the coection capability ofc v. In this wok, a coding scheme is poposed in which this excess edundancy is eliminated, esulting in signicant savings in the oveall edundancy compaed to the conventional case, while oeing the same eo potection. he edundancy of the poposed scheme is n h v + h (ln v +O(1))+ v, whee the paametes h and v ae close in value to thei countepats in the conventional case, which has edundancy n h v + n v h h v. In paticula, when the codes C h and C v have the same ate and h n h, the edundancy of the poposed scheme is close to one half of that of the conventional poduct code countepat. Vaiants of the scheme ae pesented fo channels that ae mostly busty, and fo channels with a combination of andom eos and bust eos. Keywods: aay codes, genealized concatenated codes, poduct codes, supeimposed codes. Compute Systems Laboatoy, Hewlett-Packad Laboatoies, Palo Alto, Califonia, and Hewlett-Packad Isael Science Cente, Haifa, Isael. On sabbatical leave fom the Compute Science Depatment, echnion, Haifa 32000, Isael. onny@cs.technion.ac.il. y Compute Systems Laboatoy, Hewlett-Packad Laboatoies, Palo Alto, Califonia. seoussi@hpl.hp.com. Intenal Accession Date Only

2 1 Intoduction Poduct codes [][13] ae a popula choice of eo coection mechanism in magnetic ecoding due to thei ability to oe good potection against both andom and bust eos. Figue 1 depicts a typical n v n h aay ove a eld F = GF (q) which is encoded by a poduct code consisting of two codes: an [n h ;k h =n h h ;d h ] ow code C h ove F and an [n v ;k v =n v v ;d v ] column code C v ove F. Heeafte we will efe to this poduct-code constuction as Constuction 0. he oveall edundancy of Constuction 0isgiven by n h v + n v h h v : (1) In many applications, the codes C h and C v ae taken to be maximum-distance sepaable (MDS) codes such as Reed-Solomon (RS) codes, in which case d h = h +1 and d v = v +1. his equies having code lengths n h and n v which do not exceed q+1, a condition that is met in pactice in cases whee the codes ae natually symbol- (e.g., byte-) oiented, and whee bust coection is a majo objective. heefoe, we will assume thoughout this wok that the codes used ae MDS. n h - k - - h h n v k v? v?? p p p p p p p p p p p p p p p p p p p Checks on Raw Data p Rows p p Codewod of C v p p p p p p p Checks on Columns Checks on Checks Codewod of C h Figue 1: Aay of a poduct code. In ou model of eo values, we will assume that enties in a tansmitted aay ae aected, esulting in an aay ~. An aected enty is eplaced by avalue of GF (q) which is unifomly distibuted ove the elements of GF (q), independently of the oiginal contents of o of the 1

3 othe eo values. Such a model is appoximated in pactice though the use of scambles. Notice that, in paticula, an aected enty may still keep the coect value with pobability 1=q. If its value has been changed, we say that this enty is coupted. he eo aay is dened by E = ~. he eo pattens that will be consideed in this wok ae mainly bust eos [13]. Assuming that the encoded aay is tansmitted ow by ow, then, by the natue of bust eos, we expect the aected enties in the eceived aay ~ to be conned to a numbe of ows, whee is govened by some pobability measue Pobf = t g which depends on the channel and on the choice of n h and n v (see below). An aected (espectively, coupted) ow in ~ isaow that contains at least one aected (espectively, coupted) enty. With the exception of Section, we will not assume any paticula model on the pattens of aected enties within an aected ow. If the ith ow in ~ has been aected, then the espective eo vecto is given by the ith ow of the eo aay E. An eo vecto is nonzeo if and only if the espective ow in ~ has been coupted. Atypical bust decoding stategy fo Constuction 0isas follows: he code C h is st used to detect the coupted ows in a way that we descibe shotly. Having found the coupted ows, the decode of C v is applied column by column, now egading the coupted enties in each given column as easues. If p is the acceptable pobability of aay miscoection, we will allocate half (say) of this pobability to the event that the numbe of eos exceeds the coection capability ofc v.in paticula, if C v is an MDS code, then v can be taken so that Pobf > v g p=2 : (2) his guaantees that the easue coection capability of C v is acceptable. (In fact, condition (2) can be slightly elaxed, since it is sucient to equie that, with pobability 1 (p=2), the numbe of coupted as opposed to aected ows in ~ does not exceed v.) Now, the code C h detects the coupted ows by computing, fo each ow, its syndome with espect to C h. Let ` be the numbe of aected enties in a given aected ow. If ` < d h = h +1, then the computed syndome fo that ow must be nonzeo in case the ow is coupted. Othewise, suppose that ` > h fo a given aected ow. Since evey h columns in any h n h paity-check matix of C h ae linealy independent (by vitue of C h being MDS), the pobability that such an aected ow has an all-zeo syndome is q h (futhemoe, the pobability that such a ow has got coupted in addition to having an all-zeo syndome is q h q ` < q h ). heefoe, egadless of the numbe of aected enties in a coupted ow, the pobability of misdetecting a coupted ow is less than q h. It follows that the pobability that a ow in a given aay is both coupted and misdetected is less than P t Pobf = tgtq h =q h, whee stands fo the expected value E f g. In fact, since we assume that (2) holds, then it is sucient to equie that h is such that X t Pobf = t j v gtq h p=2 ; 2

4 o ( v ) q h p=2 ; (3) whee () =Efj g(and whee we assume that h does not exceed n h ). We point out that the choice of h though (3) is athe consevative (and theefoe obust) in the sense that we equie that the oveall pobability of misdetecting a ow will be not geate than p=2. Fo instance, in the event that the numbe of aected ows is much smalle than v, we could in fact allow the decode of C h to misdetect some of the coupted ows and take advantage of the emaining v edundancy symbols (in excess of )inc v to locate the misdetected coupted ows. Such tuning, howeve, will depend much moe substantially on the behavio of the pobability measue Pobf = t g, wheeas (2) and (3) depend only on the (conditional) expected value of and the point whee the tail pobability dops below p=2. Indeed, in Appendix B we demonstate how a ne tuning of the paametes can be made though a moe extensive dependence on the pobability measue Pobf = t g. he consevative appoach, howeve, is waanted in many pactical applications whee the chaacteization of the channel statistics is often athe poo. Many vaiations on the decoding stategy of Constuction 0 ae possible, oeing a tade-o between andom and bust eo coection. he consideations fo detemining the values of n h and n v in the bust model case ae oughly as follows. On the one hand, we would like n h to be as small as possible so that the numbe of enties that will be maked as eased by the decode of C h will be close to the numbe of enties that ae aected by the busts. On the othe hand, we would like n h to be lage enough so that the atio h =n h and hence the elative edundancy beassmall as possible. Also, n v and theefoe v must be small enough so that, by the law of lage numbes, we will be able to maintain a suciently small value fo the atio v =n v while still satisfying (2). his howeve makes the decode of C v moe complex, as it needs to be able to coect moe easues. An uppe bound on n h n v is dictated by the amount of memoy and latency that we can aod. In this wok, we obseve that although Constuction 0 oes vey good eo potection, it contains excessive edundancy, due to the fact that the \inne" code C h povides the \oute" code C v with infomation on many eo pattens that exceed the coection capability ofc v. Moe specically, we allocate edundancy h of C h fo each ow of the aay to detemine whethe the ow is coupted. his way, the decode of C h can infom the decode of C v about any combination of up to n v coupted ows. Howeve, the code C v can coect only up to v eased locations, namely, it can only handle up to v coupted ows. Any infomation about combinations of v +1 coupted ows o moe is theefoe useless fo C v. Nevetheless, we ae paying in edundancy to povide this infomation. A coding scheme whee this excess edundancy is eliminated is pesented in Sections 2 and 3. Section 2 pesents a basic constuction, efeed to as Constuction 1, that illustates the key ideas and achieves most of the edundancy eduction, while Section 3 pesents a moe ened constuction, efeed to as Constuction 2, that attains futhe edundancy gains though the use of codes with vaying ates. 3

5 In the ealy wok by Kasahaa et al. [11], they suggested an impovement on Constuction 0 by a technique called supeimposition. he objective in [11] was inceasing the code dimension while maintaining the minimum Hamming distance of the code. he same motivation also led to the intoduction of genealized concatenated codes by Blokh and Zyablov in [4]. In those codes, the savings in the oveall edundancy wee obtained by using inne and oute codes with vaying ates. Genealized concatenated codes wee futhe studied by Zinoviev [18], and Zinoviev and Zyablov [19], [20], whee the latte pape also consideed minimum-distance decoding of combined andom and bust eos. Hiasawa etal.[7],[8] pesented a simila constuction which was shown to incease the code ate while maintaining the miscoection pobability of andom eos. Fo elated wok, see also [12] and [1]. Ou main objective in this pape is to incease the code dimension while maintaining the miscoection pobability of busts (we do conside also a moe geneal setting in Constuction 3 of Section that includes combined bust and andom eos). Ou constuctions die signicantly fom that of Kasahaa et al. [11] in the decoding mechanism (which we pesent in Section 4), although the schemes do bea some esemblance in thei encoding mechanisms (ou encode is pesented in Section 5). Howeve, the dieent objective allows us to obtain a moe substantial impovement on the code dimension ove Constuction 0 compaed to the constuction in [11]. Most aspects of ou constuctions also die fom those of Blokh and Zyablov [4] and Hiasawa et al. [7],[8]. Still, it is woth pointing out a featue which appeas both in those constuction and Constuction 2, namely, that of using a sequence of codes of vaying ates athe than a unique code theeby inceasing the oveall code ate while maintaining the miscoection pobability. We also mention hee the ecent wok [17], whee the model of cisscoss eos is studied. hat model is moe geneal than the one we discuss hee; howeve the constuction fo cisscoss eos equies moe edundancy. 2 Simple constuction with educed edundancy (Constuction 1) Let C h and C v be the MDS codes ove F = GF (q) which ae used in Constuction 0 and let [n h ;k h =n h h ;d h = h +1] and [n v ;k v =n v v ;d v = v +1] be thei espective paametes. Also, let H h be an h n h paity-check matix of C h. Let be an aay which consists of n h columns, each being a codewod of C v. Unlike Constuction 0, we do not assume at this point that the ows of belong to any specic code. Fo each ow of, we compute its syndome with espect to the paity-check matix H h, thus obtaining an n v h syndome aay S = S( ); that is S = H 0 h : (Note that each ow in S can take abitay values in F h.) 4

6 Now, suppose that is tansmitted though a noisy channel, esulting in a (possibly coupted) aay ~ = +Eat the eceiving end. Let S ~ be the syndome aay ~ H 0 h coesponding to the eceived aay (see Figue 2). We compae S ~ to the syndome aay S = S( ) fo n h - h - n v k v s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s Raw Data? v?? Received Aay ~ Syndome Aay ~ S Figue 2: Syndome aay fo eceived aay. the tansmitted aay. 1 Moe specically, we conside the dieential syndome aay he following two obsevations can be made: S = ~ S S = EH 0 h : If a given ow insis nonzeo (namely, if the contents of a given ow in Shas ~ changed compaed to that ow in S), then the espective ow in has been coupted. If a given ow in S is all-zeo, then the pobability that the espective ow in been coupted is less than q h. has Hence, if condition (3) holds, then, with pobability 1 point at all the coupted ows in ~. (p=2), the nonzeo ows of S 1 Fo the time being, this \compaison" is only conceptual, as the oiginal syndome aay S is not sent along with the tansmitted aay, and is not available on the decoding side. 5

7 Let S 0 ;S 1 ;...;S h 1 denote the columns of S = S( ). In ode to allow the eceive to locate the nonzeo ows in S = ~ S S, the tansmitte encodes the aw data in so that each column vecto S j in the esulting syndome aay S = S( ) is a codewod of an [n v ;n v j ; j +1] MDS code C j ove GF (q). he choice of the edundancy values j should allow the eceive to locate the nonzeo ows in S out of ~ S, with an acceptably small pobability of failue. A simple choice fo C j, which we assume fo the emainde of the section, would be setting j = 2 v fo each j, whee v satises condition (2). his condition implies that with pobability 1 (p=2), the numbe of nonzeo ows in S will not exceed v. heefoe, in this case, by decoding the columns of S, ~ the decodes of Cj can locate the nonzeo ows of S and, thus, the coupted ows of with pobability > 1 q h 1 (p=2). Now, each column vecto S j is obtained as a linea combination of the columns of. Each column of, in tun, is a codewod of C v. Since the code C v is linea, it follows that each column S j is a codewod of the MDS code C v whose edundancy is v. Howeve, we equie that S j belong to an MDS code C j with edundancy j =2 v, making the oveall edundancy in S equal to 2 h v. If we choose each C j to be a subcode of C v, then we can fully exploit the edundancy inheited fom C v due to lineaity. he equied additional edundancy of h v in S will be achieved by imposing h v additional linea constaints on the encoded aay. We efe to the esulting scheme as Constuction 1, and fom the above discussion, we eadily obtain the following. Poposition 1. he edundancy of Constuction 1 is n h v + h v : (4) Hence, the edundancy of Constuction 1 compaes vey favoably with (1) when v n v, as is usually the case in pactical applications. In paticula, when h =n h = v =n v and h ; v n h ;n v, the eduction in the edundancy is close to a facto of 2 compaed to Constuction 0. 3 Futhe edundancy eduction (Constuction 2) Additional savings in edundancy can be achieved by obseving that fo each 1 j < h, the decode of C j can obtain easue infomation fom columns of ~ S that have aleady been decoded. his will esult in a coding scheme which will be efeed to as Constuction 2. As we show next, the oveall edundancy of Constuction 2 is at most n h v + h (ln v +O(1))+ v, whee the paametes h and v ae close in value to thei countepats in Constuction 0. In Constuction 2, we encode the aay as befoe, so that each column is a codewod of the [n v ;n v v ; v +1] code C v and, fo 0 j < h, each column vecto S j in S = S( ) is a

8 codewod of an [n v ;n v j ; j +1] code C j. We will detemine the paametes v, h, and j late on. Let S0 ~ ; S1 ~ ;...; Sh ~ 1 denote the columns of the possibly coupted syndome aay ~S. We assume that the eceive decodes those columns in a consecutive ode, stating with ~S 0. Since the code C 0 does not have any a pioi easue infomation, its edundancy will be set to 0 =2 v in ode to locate up to v eos in S0 ~. As mentioned, if a ow ins= S ~ Sis nonzeo, then the espective ow in ~ is coupted (futhemoe, the convese holds with pobability > 1 q h ). Howeve, thee may be nonzeo ows in S that ae missed by the C 0 -decode: hese ae the nonzeo ows in S whose leading enty (i.e., thei enty in S 0 = S0 ~ S 0 ) is zeo. Nevetheless, with high pobability (which we compute next), most of the nonzeo ows in S will be found by the C 0 -decode when applied to S0 ~, and the locations of those ows can be passed to the decodes of C 1 ; C 2 ;...;C h 1 as easue infomation. hese decodes, in tun, can locate nonzeo ows in S that wee missed by the C 0 -decode. In geneal, fo 1 j < h, the C j 1 -decode will pass easue infomation to the C j -decode, thus allowing the eduction of the edundancy of C j which is equied in ode to decode ~S j. Ultimately, the easue infomation passed by the C h 1-decode to the C v -decode will include (with an acceptably small pobability of failue) the locations of all the coupted ows in ~, leaving the C v -decode with the task of decoding easues only. Fo the sake of unifomity, it will be convenient to dene C j and j fo j = h as C v and v, espectively. hus, we will have a gadual tansition fom full eo coection fo C 0, though combined eo{easue coection fo C j, 1 j < h, to pue easue coection fo C h = C v. he detemination of the edundancies j, 0 j h, in Constuction 2isdiscussed next. 3.1 Setting the constaints on the code paametes Let H h be the paity-check matix of C h used to compute S and, fo 1 j h, denote by H [j] h the j n h matix which consists of the st j ows of H h. We say that H h satises the MDS supecode popety if, fo j =1;2;...; h, each matix H [j] h is a paity-check matix of an MDS code. A code C h is said to satisfy the MDS supecode popety if it has a paity-check matix that satises the MDS supecode popety. Examples of matices that satisfy this popety ae H h = [ k` ] h 1;n h 1 k=0;`=0, whee the ` ae distinct elements of GF (q); these ae paity-check matices of genealized RS codes. Notice that when h > 1, evey matix that satises the MDS supecode popetymust be nonsystematic. Since evey MDS code has a systematic paity-check matix, it follows that codes that satisfy the MDS supecode popety also have paity-check matices that do not satisfy the popety. We also point out that thee ae MDS codes, such as the [q+1;q+1 ;+1] (doubly-extended) RS codes with > 1, that do not satisfy the MDS supecode popety. We elaboate moe on this in Appendix A. We will assume in Constuction 2 that H h satises the MDS supecode popety. 7

9 Conside the syndome aay S = by the st j columns of S, i.e., H 0 h. Fo 1 j h, denote by S [j] the matix fomed S [j] = [ S 0 S 1... S j 1 ] = H [j] 0 h : We dene ~ S [j] in a simila manne and we let S [j] be ~ S [j] S [j]. We say that a coupted ow in ~ is hidden fom ~ S [j] if the coesponding ow in S [j] is all-zeo. Fo 1 j h, denote by X j the andom vaiable which equals the numbe of coupted ows in ~ that ae hidden fom ~ S [j]. We extend this denition to j = 0, letting X 0 denote the numbe of coupted ows in ~. We will assume that fo 1 j < h, each code C j is a subcode of C v and so we can wite j = v + a j whee a j 0. he oveall edundancy of Constuction 2 thus equals v n h + h 1 X j=0 a j : (5) We will also dene a h =0,inaccodance with ou pevious convention that C h = C v. We can now fomulate ou poblem as follows: Given an acceptable pobability p of miscoection, nd nonnegative integes v ( n v ), h ( n h ) and a 0 ;a 1 ;...;a h 1 (and a h =0) that subject to the constaint Pob minimize v n h + n h [ j=0 X h 1 j=0 a j ; () f X 0 + X j > v +a j g o p: (7) he constaint (7) eplaces conditions (2) and (3) and guaantees, with acceptable pobability, that fo each j, the numbe of eos, X j, and easues, X 0 X j, does not exceed the coection capability of the code C j (i.e., the edundancy is at least (X 0 X j )+2X j ). In the sequel, we nd an appoximation to the solution of () and (7). By (7), we must have Pobf X 0 + X j > v +a j g=o(p)fo all j. In paticula, fo j = h we need to have Pobf X 0 + X h > v g=o(p), which, in tun, implies that Pobf X 0 > v g=o(p). On the othe hand, fo j =0weneed to have Pobf 2X 0 > v +a 0 g=o(p), which thus motivates us to choose a 0 = v. his choice is also consistent with ou stategy that the C 0 -decode handles as many as v eos without having any a pioi infomation on the locations of those eos. Now, Pob n h [ j=0 f X 0 + X j > v +a j g o 8

10 Pob n [ h f + X j > v +a j g o v + Pobf > v g j=0 X h n Pob j=1 + X j > v +a j v o + Pobf > v g; whee the second inequality follows fom a union bound and the fact that X 0 v given the conditioning event (and so the tem that coesponds to j = 0 vanishes). It thus follows that (7) is implied by Pobf > v g p=2 (8) and h X j=1 n Pob + X j > v +a o j v p=2 : (9) Satisfying the constaint (9) guaantees with acceptable pobability that each C j -decode will have enough edundancy to coect the numbe of full eos, X j and easues, X j, that it will typically encounte. he expessions Pobf + X j > v + a j j g will be bounded fom above using Lemma 1 below. Lemma 1. Fo 1 j h and evey nonnegative intege b, Pobf X j >bjg q j(b+1) b +1 ; whee t b+1 =0if b t. Poof. Suppose that ow i in ~ contains at most j aected enties and let e be the espective eo vecto. Since H h satises the MDS supecode popety, H [j] h is the paity-check matix of an [n; n j; j+1] code and, thus, ow i in S [j] is all-zeo if and only if e = 0. Hence, if at least one of the (up to) j aected enties of ow i in ~ has been coupted, then that ow cannot be hidden fom S ~ [j]. Next conside the ows in ~ that contain moe than j aected enties, and let be the andom vaiable which equals the numbe of those ows. Fo each such ow, the espective ow in S [j] will be zeo with pobability q j. Futhemoe, the vecto values of the ows in S [j] that coespond to distinct aected ows of ~ ae statistically independent. Hence, ecalling that, we have, Pobf X j >bj; g X z=b+1 b +1 9 z q jz (1 q j ) z q j(b+1) q j(b+1) ; b +1

11 and, theefoe, n o Pobf X j >bjg = E Pobf X j >bj;g q j(b+1) ; b +1 as claimed. Fo nonnegative integes, s, and, wedene the quantity B (;s;)by ( ) B(;s;) = E : (10) s+1 Clealy, B (;s;) = 0 when s 2. denition of B and fom Lemma 1. he following coollay follows eadily fom the Coollay 1. Fo 1 j h and evey nonnegative intege a, Pobf + X j > v +aj v g B( v ; v +a; q j ) q j(a+1) : By Coollay 1, we can eplace the constaint (9) by the following stonge condition, X 1j h : a j < v B ( v ; j ;q j )q j(a j+1) p=2 ; (11) whee we ecall that j = v + a j and a h =0. Notice that we have esticted the summation index set in (11) to those values of j fo which a j < v, since B ( v ; j ;q j ) = 0 othewise. In fact, if Pobf = v g > 0 (which is the case if v is the smallest intege that satises (8)), then we can also state convesely that B ( v ; j ;q j )>0 wheneve a j < v. We next show a feasible solution fo the a j, satisfying the constaint (11). his solution will be the basis of Constuction 2, since, togethe with the equiement (8), it will also satisfy constaint (7) and povide an appoximation to (). 3.2 Analysis of a feasible solution Fo a nonnegative intege, dene () by () = q Efq (2 1) j g : (12) heoem 1. Given an acceptable pobability p of miscoection, let v be a positive intege (such as an intege that satises (8)) and let h be an intege such that log q q q 1 ( v ) p=2 10 h n h (13)

12 (povided that such an intege h exists). hen the following values a j = ( v if 0 j < h = v d h =je 1 if h = v j h ; 0 j h ; (14) satisfy the constaint (11). Poof. Let j 1 <j 2 <... < j s be a sequence which consists of all indexes 0 <j h such that a j = a j 1 ; note that j s = h, and dene j s+1 = j s +1. Fix to a value less than o equal to v. Fo evey 1 ` s we have, j`+1 1 X j=j` j +1 q j( j+1 ) = q j`+1 q 1 q j`( j ` +1 j`+1 1 X j=j` ) q q 1 q h q j`( v) q j( j` +1 ) j`+1 j`+1 ; (15) whee (15) follows fom (14), namely, j`( j` +1)=j`(a j`+1+ v ) h +j` v. Hence, fo evey xed v we have, X 1j h : a j < v j +1 q j( j+1 ) = (15) sx `=1 j`+1 1 X j=j` j +1 q q 1 q h q j 1( v) q j( j+1 ) sx sx `=1 j`+1 q q 1 q h q v `=1 j`+1 q q 1 q h q v q (2 1) : aking expected values with espect to the pobability measue Pobf = t j v g yields X 1j h : a j < v B ( v ; j ;q j )q j(a j+1) whee the last inequality follows fom (13). Note that fo the values of a j dened in (14) we have q q 1 q h q v E q q 1 q h ( v ) p=2 ; v = a 0 a 1... a h 1 a h = 0; nq (2 1) v o (1) 11

13 and so fo 1 j h, the code C j 1 can be taken as a subcode of C j. In ode to compute h fom (13) we need to get uppe bounds on ( v ). We obtain such bounds in Appendix B, but we mention hee the vey simple bound () 2 1 : (17) he condition h n h will be satised if the acceptable pobability of eo p is at least 2q q 1 ( v )q n h ;by (17), this lowe bound on p is smalle than 2q q 1 2v q n h. Now, if p is smalle than this bound, we will need to take h = n h and incease v so that C v will be able to coect a cetain numbe of eos, in addition to easues (note that a simila poviso on p is also implied by (3)). his situation, though, will be faily atypical, and it will pobably mean that the initial design paametes n h, n v, o q might need to be e-thought. Remak. Inequality (1) in the poof of heoem 1 holds with equality if j 1 = 1, which occus if h v. Othewise, an impovement of the left-hand side of (13) can be obtained by intoducing the positive intege vaiable J, esulting in the following bound on h, whee n o min max (J 1) v +1; log q (q=(q 1)) ( v ;J)=(p=2) h n h ; (18) J>0 (;J) = q J E f q J (2 1) j g : (19) By (18), the minimizing J is at most j 1 = d h = v e. Lemma 2. Let h and a j be dened by (13) and (14). hen, h X j=0 a j d h = v e v + ( h 1) (ln v + ) ; whee =0: is the Eule constant [1, p. 255]. Poof. Wite j 1 = d h = v e. By (14) we have, h X j=0 a j j 1 v + X h 1 j=j 1 & h j ' X h 1 1 = j 1 v + ( h 1) j j=1 1 j 1 v + j 1 1 X j=1 j 1 v + ( h 1) (ln h + ln j 1 ) ; 1 j X h 1 h 1 j=j 1 j whee we have used the inequalities ln x P x 1 j=1(1=j) ln x +. Hence, h X j=0 a j j 1 v + ( h 1) ln ( h =j 1 )+ dh = v e v + ( h 1) (ln v + ) ; 12

14 as claimed. We summaize the foegoing discussion by bounding the edundancy of Constuction 2 in the following poposition, which follows fom (5) and Lemma 2. Poposition 2. he edundancy of Constuction 2 is bounded fom above by whee v and h ae set accoding to (8) and (13). n h v +( h 1)(ln v + +1)+ v +1; (20) 3.3 Computing the code paametes he bound (17) on () allows us to estimate the left-hand side of (13) and set h to & ' v log h = 2 (p=2) log 2 (q 1) + 1 : (21) log 2 q In compaison, the value of h fo Constuction 0 and Constuction 1, as dictated by (3), is given by & ' log2 (( v )) log h = 2 (p=2) : (22) log 2 q Hence, when ( v )=E fj v g1, the value of h in (21) is lage than the one in (22) by an additive tem which is at most d v = log 2 qe +1. heefoe, applying Constuction 0 o Constuction 1 with the consevative appoach (namely, a coding appoach whee we insist on keeping the ow misdetection pobability uppe-bounded by p=2), the edundancy (20) of Constuction 2 can be signicantly smalle than the edundancy (4) of Constuction 1 (and hence much smalle than the edundancy (1) of Constuction 0). he development leading to (21) and (22) was based, in both cases, on the consevative appoach, which assumes vey little on the behavio of the actual pobability distibution Pobf = t g. A ne computation and compaison is possible when moe detailed knowledge of the distibution is available. An example of such analysis is pesented in Appendix B fo a Benoulli distibution. Howeve, the consevative appoach is the appopiate choice in the following so-called cut-o ow-eo channel. In this channel, we assume that thee is a cut-o eo count c such that Pobf > c gp=2 and Pobf = t j c g = 8 >< >: 1 c t =0 c t= c 0 othewise he cut-o ow-eo channel models (in a athe simplied manne) a case whee the aay may be susceptible to one long bust event occuing with pobability (1 (p=2)) c, and such 13 :

15 an event aects seveal ows in the aay; in ou simplied model we assume that the bust aects exactly c ows, which makes it moe amenable to exact analysis. By (2) and (8) we can take v = c and have ( v ) = E f j v g = v c : he computation of ( v )isathe staightfowad and we obtain ( v ) = q v E f q (2 1) j v g = (2 v 1) c : Hence, by (13) we can choose & ' v + log h = 2 c log 2 (p=2) log 2 (q 1) + 1 log 2 q & v log = 2 v + log 2 (( v )) log 2 (p=2) log 2 (q 1) log 2 q ' + 1 (23) (note that this value can be smalle than the one in (21)). On the othe hand, it is easy to veify that, fo this channel, (22) povides the ight choice fo h fo Constuction 0 and Constuction 1. It is also easy to check that the edundancy (20) of Constuction 2 can thus be signicantly smalle than the edundancy (4) of Constuction 1 fo this channel. We illustate this in the next numeical example. Example. Conside the design of a code with n h = 9, n v = 128 ove F = GF (2 8 ), with a taget aay eo ate of p = We assume a cut-o ow-eo model as descibed above, with c =10 3 and c = 10. We set v = c =10and by (23) we take h =8. Next, we use (14) to obtain a 0 =10;a 1 =7;a 2 =3;a 3 =2;a 4 =a 5 =a =a 7 =1;a 8 =0. he total edundancy is n h v + P j a j = 98 symbols. In compaison, fo Constuction 0 and Constuction 1, we take by (22) h = 7, esulting in a edundancy of 1030 symbols fo the latte and a edundancy of 178 symbols fo Constuction 0. In Appendix B, we analyze the Benoulli ow-eo channel. In this channel, each ow gets aected with pobability = =n v, independently of the othe ows. It tuns out that fo typical values of q, n v, and we can take & ' v + log h = 2 ( v +1) log : (24) log 2 q On the othe hand, in Constuction 0 and Constuction 1weneed to take & 3 2 h = log ' 2( v +1) log 2 + O(1) ; (25) log 2 q 14

16 and such a value of h is equied also if we do not insist on the consevative appoach. Hence, the edundancy of Constuction 2 will be typically smalle than that of Constuction 1 fo the Benoulli ow-eo channel, and theefoe typically much smalle than that of Constuction 0. Remak. When compaing ou constuction with Constuction 0, we have chosen a delity citeion which is the pobability p of having a miscoection in any given n v n h aay. he pefomance of a coding scheme can be measued also in tems of the eective `symbol eo pobability' afte decoding, which equals the aveage faction of eoneous enties among the decoded enties. Fo Constuction 0, the stategy that we have outlined used the code C h fo detection only. heefoe, C h will neve miscoect, namely, it will neve identify an unaected ow as coupted (on the othe hand, it might misdetect coupted ows). heefoe, the dominant failue event fo these codes is one in which v +1 ows ae coupted, and the condition is detected by C h, which pevents C v fom doing any futhe \damage." Unde the constaint (8), the eective symbol eo pobability afte decoding in this case is appoximately (p=2)( v +1)=n v. In Constuction 2, howeve, the codes C j might miscoect; by the constaint (9), this will happen with pobability p=2 wheneve v. Such a miscoection, in tun, might intoduce up to v false coupted ows in the decoded aay, amounting to an incease of (p=2) v =n v in the eective symbol eo pobability afte decoding. o esolve this, we need to choose a value fo p which is one half of the value chosen in Constuction 0. Since the dependence of the paametes on p is logaithmic, such achoice of p has a small (if any) eect on the code paametes. 3.4 Summay of Constuction 2 o summaize, Constuction 2 is obtained as follows: Given n h, n v, and p, set the paamete v that (8) holds. to be the smallest positive intege such Set the paamete h so that (13) holds. Set the code C h to be an [n h ;n h h ; h +1] code ove F with an h n h paity-check matix H h which satises the MDS supecode popety. Fo 0 j h, set C j to be an [n v ;n v j ; j +1] code ove F such that j = v + a j and a j is given by (14). Futhemoe, each code C j 1 is a subcode of C j : the j n v paity-check matix H j of C j consists of the st j ows of the j 1 n v paity-check matix H j 1 of C j 1. We let C v and H v be C h and H h, espectively. Let h 0 ; h 1 ;...;h h an n v n h aay 1 denote the ows of H h. In Constuction 2, the aw data is encoded into such that the following holds: 15

17 Fo 0 j < h, H j h 0 j=0; (2) namely, when the syndome of each ow of is computed with espect to the paitycheck matix H h, an n v h aay S =[S 0 S 1... S h 1]= Hhis 0 obtained in which each column S j is a codewod of C j. Each column in isa codewod of C v = C h, namely H v = 0 : (27) We can ewite (2) and (27) as H j H [j+1] 0 h =0; 0jh ; whee H [j+1] h is the matix which consists of the st j+1 ows of H h and H [ h+1] h as the n h n h identity matix. is dened 4 Decoding he decoding pocedue of Constuction 2 can be summaized as follows. Let be the tansmitted aay and let ~ be the eceived aay. Fo each j, 0 j h 1, j will denote the numbe of eased locations input to the decode of C j fom pevious stages. 1. Compute the syndome aay ~ S =[ ~ S0 ~ S1... ~ Sh 1]=~ H 0 h. 2. Set 0 =0. Fo j =0;1;...; h 1, do a Given the locations of j easues in column ~ Sj, apply an eo-easue-coecting decode fo C j to locate up to b( v +a j j )=2c additional full eos in column ~ Sj. Fo each full eo found in ~ Sj, mak the coesponding ows in ~ S and ~ as eased. b Let j be the numbe of full eos found in Step 2a. Set j+1 = j + j. c If j+1 > v,declae the aay undecodable and stop. 3. Apply an easue-coecting decode fo C v to ecove a total numbe of up to v easues in each column of ~. We point out that the pobability of miscoection will be bounded fom above by p also if we limit the numbe of full eos that we attempt to coect in Step 2a to minfa j ; b( v +a j j )=2cg. Steps 2 and 3 can be implemented by choosing the codes C j to be RS codes and using any of the known decoding algoithms fo these codes, designed to handle both eos and easues. 1

18 he basis of those algoithms is computing an eo-locato polynomial (z) ove the eld F (see [2, Ch. 7], [3], [15]) in an iteative manne, such that the oots in F of the computed polynomial (z) indicate whee the locations of the eos ae. If the locations of some eos ae initially known (i.e., if some of the eos ae actually easues), then this infomation can be incopoated into the RS decoding algoithm by a pope initialization of the polynomial (z). In the aay decoding pocedue outlined above, each stage j, 0 j h 1, poduces an eo locato polynomial j, which is then fed to the next stage as the initial value of its eo locato polynomial j+1. Moe specically, we st compute an eo-locato polynomial 0 (z) fo the column S0 ~ of S. ~ By Lemma 1, the pobability of having any coupted ow which is hidden fom S ~ [1] =[~ S0 ]satises Pobf X 1 > 0 g =q (whee = E f g). Since the edundancy of C 0 is 2 v,wewill expeience a decoding failue on ~ S0 only when the numbe of coupted ows in ~ exceeds v. Indeed, the constaint (8) guaantees that this will occu only with an acceptably small pobability. Fo subsequent columns of ~ S, we compute an eo-locato polynomial j (z) that points at the eoneous ows in ~ S [j+1] = [ ~ S0 ~ S1... ~ Sj ]. he C j -decode will fail on the column ~ Sj only when fo some ` j, the numbe of coupted ows that wee hidden fom ~ S [`] exceeds the coection capability ofc`; this occus when X 0 + X` >`. Howeve, the constaints (8) and (9) (which imply (7)) guaantee that this pobability is acceptably small fo all j. By Lemma 1, the pobability that C j will need to coect pope eos (in addition to easues) satises Pobf X j > 0 g q j : We can theefoe conclude that when v n v q, most of the eo-locating eot will typically fall on the C 0 -decode while computing the eo-locato polynomial 0 (z). he ole of the est of the columns of S ~ amounts, in most cases, to veifying, with an acceptably small pobability of eo, that 0 (z) is the tue eo-locato polynomial. If the polynomial 0 (z) tuns out to be inconsistent with any of the subsequent columns in S, ~ then it will be updated by the decoding algoithm when applied to those columns. At any ate, by well-known popeties of linea-ecuing sequences [15], it can be shown that the numbe of such updates is bounded fom above by the numbe of actual coupted ows, assuming that no failue has occued in the decoding of any of the columns of S. ~ hus, the total numbe of opeations pefomed in atypical execution of the aay decoding pocedue will be signicantly smalle than the numbe of opeations in h independent RS decodings. 5 Encoding In this section, we outline an encoding pocedue fo Constuction 2. he encode descibed hee esembles the one in [11], with the following two majo dieences: 17

19 he new encode is systematic, namely, the aw data is included, as is, in the encoded aay. he encode in [11], on the othe hand, encodes pat of the data non-systematically. he new encode is moe geneal in the sense that the codes C j have dieent edundancies. he aw data is assumed to be enteed into column by column, stating at the column n h 1 and ending with 0. We denote the esulting evesed aay by (. We beak the encoding pocedue into two main steps: Step A: Encoding aw data into the subaay A = [ h h n h 1] of and computing an n v v edundancy aay V =[V 0 V 1... V h 1]. Step B: Encoding the emaining pat of the aw data into the subaay B = [ h 1] though the computation of an n v v syndome aay S =[S 0 S 1... S h 1]. Step B makes use of the edundancy aay V that is computed in Step A. he computation of the columns of V can be caied out on-line while eading the data into B. heefoe, no latency will be caused duing encoding. he aays A, B, and V will be geneated in evese fom. he evesed aays will be denoted by ( A, ( B, and V (. 5.1 Encoding Step A he computation of the columns of the subaay out as follows: A and the edundancy aay V is caied Step A1: Fo j = n h 1 ;n h 2 ;...; h, inset the aw data into the st n v v enties of j. Step A2: Fo j = n h 1 ;n h 2 ;...; h, set the last v enties of j so that j becomes a codewod of C v = C h. Steps A1 and A2 ae inteleaved, and they amount to applying a conventional RS encode to obtain each column of A. 18

20 Step A3: Set the enties of V so that each ow of [ V j A ] is a codewod of C h. he computation of V can be done though accumulation of edundancy symbols while eading the data into A. Step A2 guaantees that H v A =0,inaccodance with (27). By Step A3 we have [ V j A ](H h) 0 =0 (28) fo any paity check matix Hh of C h (in paticula, the matix used hee does not have to satisfy the MDS supecode popety dened in Section 3.1). Hence, Step A3 can be easily implemented using a systematic paity-check matix of C h. In this case, the edundancy aay V can be computed column by column, while eading the data into B. 5.2 Encoding Step B Encoding Step B does depend on the specic choice of the paity-check matix H h of C h. In paticula, H h will need to satisfy the MDS supecode popety, namely, fo 1 j h, the matix H [j] h is a paity-check matix of an MDS code. Let H h = [ h k;` ] h 1;n h 1 k=0;`=0 be such a paity-check matix. Now, fo encoding puposes, we usually pefe to have matices that ae systematic (and, indeed, we did choose a systematic matix in Step A). Howeve, when h > 1, matices that satisfy the MDS supecode popety must be nonsystematic. Hence, we will equie instead the weake condition h k;` = 0 fo 0 ` < k < v and h k;k = 1 fo 0 k < h. We will efe to such a paity-check matix as uppe-tiangula (boowing the tem fom squae matices). Notice that fo each j, the st j ows of such an H h geneate an [n h ;j;n h j+1] MDS code [14, Ch. 11]; hence, fo any uppe-tiangula paity-check matix H h that satises the MDS supecode popety, we must have h k;` = 0 fo `>k. he following lemma is an immediate consequence of the fact that evey [n; n 0 ; 0 +1] MDS code has a minimum-weight codewod with zeoes in the st n 0 1 coodinates. Lemma 3. Let C be an [n; n ;+1] code ove F = GF (q) that satises the MDS supecode popety. hen C has an n uppe-tiangula paity-check matix that satises the MDS supecode popety. We explain next how the columns of B ae encoded. Let H h be an uppe-tiangula paitycheck matix of C h and such that H h satises the MDS supecode popety. Let Q = [ Q j;` ] h 1 j;`=0 be the invese of the matix fomed by the st h columns of H h. Clealy, Q is an h h uppe-tiangula matix. Now, = [ B j A ] = [ B V j0] + [V j A ]: 19

21 By (28) we can expess the syndome aay S = S( ) in the following manne: S = H 0 h = [ B V j0]h 0 h : Hence, B = SQ 0 + V ; o j = S j + X h 1 `=j+1 Q j;`s` + V j ; 0 j < h : (29) Letting () i denote the ith component of a vecto, we can ewite (29) in scala notation as follows: whee 0 i<n v. ( j ) i = (S j ) i + X h 1 `=j+1 Q j;`(s`) i + (V j ) i ; 0 j < h ; (30) Suppose that S` is known to the encode fo j <`< h. he encode computes using (30) as follows: j and S j Step B1: Wite the aw data into the st n v j enties in j. Step B2: Set the st n v j enties in S j so that (30) holds fo 0 i<n v j. Step B3: Set the last j values in S j so that S j becomes a codewod of C j. Step B4: Set the last j values in j so that (30) holds fo n v j i<n v. Steps B1 though B4 guaantee the following two popeties: (a) its st n v j enties consist of aw data, and (b) S j 2C j. j is systematic, namely, he encoding pocedue is descibed in Figue 3 in tems of the potions of the aay that ae computed in each encoding step. An auxiliay n v h aay is added fo the computation of the syndome aay S. he edundancy aay V, on the othe hand, can be computed in the same aea whee B is witten. he dotted line sepaates between the aw data and the edundancy symbols. he encoding steps that ae applied in the computation of each paticula aea of the aay ae indicated in paentheses. Applying ow eo-coection (Constuction 3) So fa in the constuctions, we have used only the detection capability of the code C h to mak the coupted ows. his is a consequence of the fact that we did not assume any model on the numbe of aected enties in a ow, thus assuming in eect the wost-case scenaio 20

22 n h - k h - h - h - Aay ( V n v k v Subaay ( A (A1) (A3) and Subaay ( B (B1) Columns S j (B2)? v Checks on Columns of ( A (A2) (B4) v? Checks on S j?? 7 So nh 1 h 1 S h 1 h 0 (B3) S 0 Aay ( Syndome Aay S = S( ) Figue 3: Encoding pocedue of Constuction 2. whee all the enties in an aected ow may get coupted. Indeed, in such a wost-case event, thee is eally no use in attempting to coect eos along ows. In this section, we incopoate patial knowledge on the distibution of the numbe of aected enties inaow and extend Constuction 2 to include some eo coection (on top of eo detection) on the ows. his appoach may be advantageous in cases whee thee is a signicant pobability to have only a limited numbe of aected enties in one ow. his is typically the case whee the channel insets both bust and andom eos. he esulting extended coding scheme will be efeed to as Constuction 3. We intoduce a design paamete,, which maks the numbe of eos that C h will attempt to coect. he ultimate design should optimize ove that paamete. he paamete will be implicit in all fothcoming notations. he andom vaiable will stand fo the numbe of aected ows each containing no moe than aected enties. he andom vaiable + will denote the numbe of ows that contain moe than aected enties. Clealy, = + +. As befoe, each column in the aay will be a codewod of an [n v ;n v v ] code C v, except that hee we set v so that Pobf + > v g p=4 : (31) 21

23 he easoning hee is that the code C v will need to coect easues only in those ows that contain moe than aected (athe, coupted) enties. We will intoduce anothe paamete, v, 0 which stands fo the oveall numbe of aected ows that Constuction 3 should be able to handle. he paamete v 0 will be detemined by the inequality which is the analog of (2) o (8). Pobf > 0 vg p=4 ; (32) he code C h is chosen to be an [n h ;n h h ]codethat satises the MDS supecode popety, whee h is set so that C h can coect any patten of up to full eos o less and detect, with suciently high pobability, any patten of moe than eos. Assuming that the decode indeed attempts to coect up to eos in each ow, the pobability that a ow containing moe than coupted enties will be misdetected o miscoected by C h is bounded fom above by q h X n h (q 1) i q h+2 ; i i=0 whee the inequality holds wheneve n h q (we show in Appendix A that this is always the case when the MDS supecode popety holds and h > 1). his bound on the pobability takes into account the wost-case scenaio whee all n h enties in that ow may get aected. Given a value of +, a decoding failue will occu only if the numbe of ows that wee miscoected by C h exceeds h + ; the pobability of this to happen is bounded fom above by + v+1 + q ( h +2)( v+1 + ). o guaantee the acceptably small pobability of decoding failue, we equie that h is chosen so that o, equivalently, E + ( (ecall the denition in (10)). + q ( h+2)( v+1 + ) v +1 + ) + v p=4 ; B +( v; v ;q h 2 )q h+2 p=4 (33) he idea behind Constuction 3 is that we codeagiven n v n h aay inaway that makes the espective n v h syndome aay S( ) a mini-aay in which we can ecove up to v 0 aected ows using the decode of Constuction 2 (when designed fo n v h aays). hose aected ows ae in fact the syndome vectos of the ows of with espect to the code C h. Now, the ows of S( ) ae aleady `scambled' vesions of the ows of though the use of the code C h. heefoe, thee will be no need to intoduce anothe ow-code (i.e., an analog of C h ) fo the ows of the mini-aay S( ). We will, howeve, need to dene a paamete h 0 and codes C 0 0 ; C0 1 ;...;C0 0 h 1;C0 = C 0 0 v that will be applied to the columns of S( ) as follows: h he codes Cj, 0 j = 0; 1;...;h 0 1, will be applied to h 0 1 columns of S( ) (say, the last columns), and C 0 v will be applied to the emaining columns. Note that 0 h is not an actual edundancy of any code in this scheme. 22

24 Following heoem 1, we set 0 h so that log q q q 1 (v) 0 p=4 0 h h ; (34) whee () is as in (12), with = + +. (If 0 h > h, then h should be inceased to have h = h.) 0 Each code Cj 0 is an [n v ;n v j] 0 MDS code whee j 0 = v 0 + a 0 j and a 0 j is given by ( 0 a 0 v if 0 j < j = 0 h =0 v dh 0 =je 1 if ; 0 j 0 0 h =0 v j h 0 h : (35) he oveall edundancy of Constuction 3 equals v n h +( 0 v v ) h + X 0 h 1 (compae with (5)), and this edundancy should be minimized ove. Given (by (32)) that the numbe of aected ows is 0 v o less, it follows fom heoem 1 that the pobability of failing to decode the syndome aay S( ) is bounded fom above by p=4. Note that the oveall pobability of the `bad events' in (31), (32), and (33), does not exceed 3p=4. Figue 4 illustates the stuctue of a coded aay in which C h is enhanced to coect eos, whee the aea below the dotted line epesents the edundancy symbols. Encoding is caied out in a manne which is simila to the desciption in Section 5. In fact, the encoding algoithm is exactly the same if we dene the codes C 0 ; C 1 ;...;C h as follows: C j = 8 >< >: j=0 a 0 j ; Cj 0 if 0 j < 0 h C 0 v if 0 h j < h : C v if j = h At the decoding side, we poceed as follows: We use the decodes of C 0 ; C 1 ;...;C h 1 to ecove the dieential syndome aay S fo the eceived aay ~. Howeve, unlike the decoding pocedue in Section 4, we do need hee to ecove the full contents of S and not just the locations of the nonzeo ows; this is done though the iteative computation of an eo-evaluato polynomial (z) fo each column of S, togethe with the eo-locato polynomial (z). Once we have the aay S, we egad each ow in S as a syndome and apply the decode of C h to attempt to coect up to eos in the espective ow in~. Decoding will succeed if thee ae at most coupted enties in that ow in~. If thee ae moe, then, by (33), the decode will detect that with suciently high pobability and mak that ow as an easue. he easues will then be ecoveed by C v. 23

25 n h - k - h - h k v n v? v?? - 0 h 0 v? 0 v? Appendix A Figue 4: Aay stuctue in Constuction 3. We summaize hee seveal popeties of codes that satisfy the MDS supecode popety. Recall that a linea [n; n ] codecsatises the MDS supecode popety if thee exist codes C = C C 1...C 0 =F n (3) such that each C j is a linea [n; n j] MDS code. We st make a connection between such codes and coveing adius [14, p. 172]. We denote the coveing adius of a code C by (C). Lemma 4. (he Supecode Lemma [5, Poposition 1].) Let C 1 and C 2 be two distinct codes such that C 1 C 2 and let d 2 be the minimum Hamming distance of C 2. hen (C 1 ) d 2. Coollay 2. Let C be a linea [n; n ] MDS code. hen (C) =if C is a subcode of some linea [n; n +1] MDS code, and (C) <othewise. Poof. Fist, it is well-known [5, Poposition 2] that the coveing adius of evey linea [n; n ] codeis at most. his, with Lemma 4, implies the st pat of the coollay. 24

26 On the othe hand, suppose that C is not contained in any linea [n; n +1;] code. hen, fo evey vecto e of length n, the coset C + e contains a wod of Hamming weight less than. his, in tun, implies that (C) <. It follows fom Coollay 2 that a linea [n; n ] MDS code C satises the MDS supecode popety if and only if all codes in the chain (3) except C 0 have coveing adii that equal thei edundancy; as such, each of these codes has the lagest (and theefoe the wost) coveing adius among all codes with the same length and dimension. A linea [n; n ] MDS code C is called maximum-length if adding any column to a paitycheck ofcesults in a paity-check matix of a linea [n+1;n+1 ] code which is not MDS. Maximum-length MDS codes ae extensively studied in pojective geomety ove nite elds, whee they ae called complete acs [9],[10]. We next make a connection between the MDS supecode popety and maximum-length MDS codes. Lemma 5. A linea [n; n ] MDS code C is maximum-length if and only if (C) <. Poof. Let C be an [n; n ] MDS code ove a eld F and let H be an n paity-check matix of C ove F. Clealy, (C) <if and only if evey (syndome) vecto h 2 F can be obtained as a linea combination ove F of less than columns in H. On the othe hand, [ H h ] is a paity-check of an [n+1;n+1 ] MDS code if and only if h cannot be obtained as a linea combination of columns in H. Combining Coollay 2 and Lemma 5, we conclude that a linea [n; n ] MDS code C satises the MDS supecode popety if and only thee is no code in the chain (3) (except C 0 ) which is maximum-length. It is known that evey linea [q+1;q 1] MDS code ove GF (q) is maximum-length [14, Ch. 11]. Hence, thee exist linea [n; n ] MDS codes ove GF (q) that satisfy the MDS supecode popety only if n q o 1. he equality n = q can be attained by extended Reed-Solomon codes. Appendix B We analyze hee the Benoulli ow-eo channel that was consideed in Section 3.3. We st impove on the value of h in (21) by obtaining bounds on () which ae tighte than (17). We do this next using the well-known Cheno bounding technique. Let x 7 U(x) be the step function which equals 1 when x 0 and equals 0 othewise. Clealy, U(x) x fo evey 0 <1. Recalling the denition of () in (12), we have () = q E f q (2 1) j g 25