Table Of Content

Multi-Block Interleaved Codes for Local and Global Read Access Yuval Cassuto∗, Evyatar Hemo∗, Sven Puchinger† and Martin Bossert† ∗Viterbi Electrical Engineering Department, Technion – Israel Institute of Technology, Israel †Institute of Communications Engineering, Ulm University, Germany [email protected],[email protected], {sven.puchinger,martin.bossert}@uni-ulm.de Abstract—We define multi-block interleaved codes as codes Feature2takescareofthefast-accessrequirement,andfeature 7 that allow reading information from either a small sub-block or 3 improves reliability with only a minor adjustment of read 1 from a larger full block. The former offers faster access, while performance in rare unfortunate instances. Note that focusing 0 the latter provides better reliability. We specify the correction on random sub-unit read performance is in line with real 2 capability of the sub-block code through its gap t from optimal minimumdistance,andlooktohavefull-blockminimumdistance applications that are much more sensitive to delayed reads n that grows with the parameter t. We construct two families of than writes. a suchcodeswhenthenumberofsub-blocksis3.Thecodesmatch The formal model that fits the above three features now J the distance properties of known integrated-interleaving codes, follows. Define a write block as N =mn symbols, where m is 5 but with the added feature of mapping the same number of 2 information symbols to each sub-block. As such, they are the the number of sub-blocks and n is the number of symbols in first codesthatprovideread accessin multiplesizegranularities each sub-block. A write unit of K =mk information symbols ] and correction capabilities. divides into m sub-units of size k each. In the write path, T K information symbols are encoded into N code symbols I . I. Introduction and written to the memory. In the read path, we need to s c return a sub-unit of k symbols by reading a sub-block of n Thetwo centralfeaturessoughtindata-storageapplications [ symbols. This operation is divided into decoding and reverse are extreme data reliability and fast data access. In data mapping, where the former corrects the errors/erasures in the 1 reliability we wish to avoid data loss in all conceivable v nsub-blocksymbols,andthelatterretrievesthek information circumstances, and in fast data access we want to read data 4 symbols from the corrected sub-block. If sub-block decoding with high throughput and low latency. Access considerations 8 failed to correct the errors/erasures, we allow reading the 1 often dictate implementing an error-correcting code over a entire write block and decode it as [N,K] code. We define an 7 fixed (and not too large) data unit, which degrades the cod- hN,K,mi multi-block interleaved code as a code supporting 0 ing performance and the resulting reliability. An especially these operationswith the parametersN,K,m. Ourobjectivein . 1 interesting instance of this happens in Flash storage, where a this paper is to construct hN,K,mi codes with good distance 0 group of multi-level memory cells is divided to smaller data properties for both individual sub-block decoding and full 7 units called pages. write-block decoding. 1 It is well known that coding over large block lengths gives : The first place to look for multi-block interleaved codes is v the best reliability for a given coding rate. But adding access within the literature on integrated-interleaving(I-I) codes[6], i performance to the considerations, block lengths are tightly X [5] (see also [1]), which provide one minimum distance for constrained by the granularity required for the read/write r sub-block codewords and a larger one for full codewords. a interface of the storage device. The standard approach of the While I-I codes give good (sometimes provably optimal) storage industry is to fix the coding block length to be the tradeoffsbetweensub-blockandfull-codeworddistances,they basic access unit of the device (e.g. a page of a few KB), cannot be readily used as multi-block interleaved codes. The and come up with the best code for this block length. A issue with I-I codes is that it is not known how to reverse- conflict between coding efficiency and access performance map a sub-block codeword to k information symbols. The thusemergeswheneverthestorageisrequiredtodeliverdataat encoder presented for I-I codes [1] maps different numbers fine accessgranularitiesthatimplyshortcodeblocks.Solving of information symbols to different sub-blocks, and it is not this conflict needs a multi-level read/write scheme with the clearhowtore-organizetheresultingmk×mngeneratormatrix following features: togetmfull-rankk×nsub-matrices.Becauseofthat,I-Icodes 1) Jointly writing m data (sub-)units(e.g. pages), each with fail to support the critical feature 2 in the read/write model. k symbols, to a single write block. Anotherrelatedcodingframeworkislocallyrecoverablecodes 2) Allowingrandomreadaccesstoasub-unit,i.e.,returning (LRC) [4], [7], [10]. LRC arefocusedonefficientlocalrepair its k symbols by physically reading a sub-block 1/m of individual code symbols, while here we are interested in smaller than the write block. local decoding of larger sub-blocks. The primary reason why 3) In cases where a sub-unit read fails due to excess errors knownLRCsarenotdirectlyapplicabletotheproposedmodel in the sub-block,readingthe fullwrite blocksucceedsin is that they partition the code coordinates to disjoint sets that retrieving the data. areeachanMDSlocalcode.Makingeachofthemsub-blocks an[n,k]MDSlocalcodedegeneratestheproblem,becauseall code, which we denote C′. Next we define the distances of parity symbols are local in this case. multi-block interleaved codes. Our main contributions in the paper are two code con- Definition2.[sub-block minimum distance] An hN,K,mi structions of multi-block interleaved codes providing flexible multi-blockcodeChassub-blockminimumdistanceδifthe tradeoffs between sub-block and full-codeword correction ca- sub-blockprojectedcodeC′hasminimumdistanceδ. pabilities.Theflexibilityisachievedbyintroducinganinteger The(full-block)minimumdistanceofanhN,K,mimulti-block parameter t specifying the gap of the sub-block code from code will be defined to be the usual minimum distance as minimum-distance optimality. As t grows, the constructed [N,K] block code. codesenjoybetter minimumdistancesfor the fullwrite-block codewords. This new view of the local vs. global correction A. Boundonthe minimumdistance ofmulti-blockinterleaved tradeoff through the t parameter allows their explicit joint- codes optimization, while in I-I codes this relation is much less Thefollowingisarestatementofaknowntheoremfrom[5], transparent (in I-I codes there is no notion of sub-block but posed in different notation (using the t parameter) that is dimension k, so t is not even defined). One family of codes, helpful for the constructions in the sequel. presentedin Section III, allowsimprovementin the full-block Theorem1. LetC beanhN = mn,K = mk,mi multi-block minimum distance (by growingt) reaching d =1.5(n−k+1). interleavedcodewithsub-blockminimumdistanceδ=n−k− An improvedconstruction, presented in Section IV, can grow t+1.Ift 6 (k−1)/(m−1)thentheminimumdistanceofCis thefull-blockminimumdistancefurtheruntild =1.6(n−k+1). boundedby The proposed constructions are given for m = 3 sub-blocks, d 6n−k+(m−1)t+1. (1) but can be extended to additional m with similar techniques. An optimal multi-block interleaved code is one meeting the The key idea toward having sub-blockreverse mappingis the bound(1)with equality.We notetwo interestingspecialcases specification of the codes through their generator matrices. of Theorem 1. For m = 1 we get the usual Singleton bound; This enables to carefully populate the generator matrix with for m=2 we get d6n−k+t+1 when t6k−1. The upper constituent Reed-Solomon (RS) generators such that all dis- bound reveals the tradeoff between the sub-block minimum tance properties are fulfilled simultaneously. Coincidentally, distanceandtheminimumdistanceofthecode.Theparameter both constructions achieve the same distance properties as t prescribesthegapofthesub-blockdistancefromoptimality, known I-I codes: the former meets 2-level I-I and the latter and in return increasing t improves the upper bound on the meets 3-level I-I. However, the constructed codes are not distance of the full-block code. reformulations of these known I-I codes: the resulting codes are different,andthere isno apparenttransformationfromthe B. Short review of Reed-Solomon codes I-I codes to our codes that adds the desired sub-block access The key building block for our constructions is Reed- features. A previous use of constituent RS generators was Solomon (RS) codes, which are now defined. in [3] to construct partial unit memory (PUM) convolutional Definition3.[Reed-Solomoncode]Letα beanelementofa codes [9], [8] (see also [2]). finitefieldGF(q)ofordern.Thendefinea[n,ℓ]Reed-Solomon II. DefinitionsandNotations codeasallpolynomialsc(x)obtainedas Throughout the paper we denote the set {a,a + 1,...,b} c(x)=i(x)g(x), by [a : b]. Similarly, [a : b]n denotes the same set, but wherei(x)isanarbitraryinformationpolynomialofdegree6 with element indices taken modulo n. For example, if n = 7, ℓ−1,and then [5 : 9] = [5 : 2] = {5,6,0,1,2}. The operation n n A \ B represents set difference between A and B. We use g(x)=(x−α−s)(x−α−(s+1))···(x−α−(s+n−ℓ−1)), (2) the term distance to refer to the Hamming distance between andsissomeintegerin[0:n−1]. vectors over the field GF(q). For notational convenience, the In simple words, an RS code is obtained by a generatorpoly- statementsinthepaperoncodeminimumdistancesarewritten nomialg(x)withn−ℓrootswhosereciprocalsareconsecutive as some d equalsomenumber,eventhoughin some caseswe powersof α. As a usefulcompactnotation,givenn andq, we only prove (the important part) that d is at least that number. define a RS code through its (cyclically) consecutive power We first define our principal object of study, which we call set. In this representation, the code defined by the generator multi-block interleaved code. in (2) is Definition1.[multi-blockinterleavedcode]An [N,K] linear RS([s: s+r−1] ), n codeCisahN,K,mimulti-blockinterleavedcodeifitscode where r = n−ℓ is called the redundancy of the code. It is coordinatesare partitionedto m disjoint sub-blocksof size well known that the minimum distance of RS([s: s+r−1] ) n = N/m each,andfromeachsub-blockwecanrecovera n is r+1. It will often be convenient to represent an RS code disjointsize-k= K/msub-unitofinformationsymbols. by the complement of its root power set, that is, For decoding a multi-block interleaved code we assume m RS([a:b] ),RS([0:n−1] \[a:b] ). projectionsofthecoordinates[1:N]todisjointsubsets[1:n], n n n [n+1 : 2n],... up to [(m−1)n+1 : mn]. We assume that the The dimension of the code RS([a : a+ℓ−1] ) is ℓ, and its n projectionofC ontoanyofthese subsetsgivesthesame [n,ℓ] minimum distance is n−ℓ+1. III. ConstructionofMulti-BlockInterleavedCodes the dimension of C is 3k, and its length is 3n. We defer the 3 discussion on encoding and decoding till after we prove the Our focus in the paper is on codes with m = 3, because minimum distances in the following. this is the most useful case for Flash-based storage devices with 8 = 23 representation levels. The constructions can be Proposition2. When t < k/2, C hassub-blockminimum 3 generalized to larger m values. For the specific case of m=3 distanceδ=n−k−t+1. we add the following definitions. Proof:Lookingatanylength-nsub-block,weobservefrom Definition4.[Di-minimum-distances] For i∈{1,2,3}, an the structure of G that the codeword projected on the sub- hN,K,3i multi-blockinterleaved code C has Di-minimum- block is a linear combination of codewords generated by distancediifthelowestweightofanycodewordwithexactlyi G1,G2,GI,GE. Noting the root power sets of these codes non-zerosub-blocksisdi. (the complements of the sets listed in Construction 1), we get: G1 → RS([t : n − 1] ), G2 → RS([2t : t − 1] ), Note that according to Definition 4, the minimum distance n n of a hN,K,3i code is mini∈{1,2,3}di. In addition to serving GI → RS([k : 2t − 1]n), and GE → RS([k + t : k − 1]n). If a power index lies in the intersection of these sets, then as upper bounds on the code minimum distance, the D- i every polynomial of a codeword in the sub-block code has minimum-distances have important operational meanings: the this power index as root. Taking the intersection of these sets D -minimum-distance gives the correction capability when 1 1 sub-blocksuffersmany errors/erasures,and the D -minimum- we obtain that the projected sub-block code is a 2 distance does the same when 2 sub-blocks suffer many er- RS([k+t:n−1] ) n rors/erasures. code, and thus its minimum distance is n−k−t+1. Construction1.LetC bedefinedbyageneratormatrixG = 3 Given the sub-block minimum distance of Proposition 2, the  G  1  GG2 ,whereG1,G2andG3arethek×3nmatricesgivenin tnieoxnt1lemismoaptsimhoawlswtihthatrtehsepDec1t-mtointhimeubmou-dnids1taonfceTohfeoCroenmstr1u.c- 3 Fig.1.NowwedefinethecomponentmatricesofFig.1;blank Lemma3.AcodeC fromConstruction1hasD -minimum- 3 1 rectanglesrepresentall-zerosub-matrices. distanced =n−k+2t+1,forallt<k/2. 1 • G1isageneratormatrixforthecodeRS([0:t−1]n). Proof: Denote the length 3k information vector as v, and • G2isageneratormatrixforthecodeRS([t:2t−1]n). partition it as • GIisageneratormatrixforthecodeRS([2t:k−1]n). v=(v ,v ,v ). • GEisageneratormatrixforthecodeRS([k:k+t−1]n). 1 2 3 Each sub-vectorv has length k, and we further partition it to j n n n v =(v ,v ,v ). j j,I j,1 j,2 k−2t GI The lengths of the m = 3 constituent vectors are from left to G = 1 right: k −2t,t,t. Consider a codeword of C with one non- t G1 GE 3 t G2 GE zero sub-block. Assume (wlog due to symmetry) that the left sub-block is non-zero. It is clear from the structure of G in Fig. 1 that all sub-vectorsexcept v must be all-zero. In this 1,I k−2t GI case, the codeword weight is at least the minimum weight of a non-zero codeword from RS([2t:k−1] ) (spanned by GI), G = n 2 t GE G1 which is n−k+2t+1 as required by the lemma statement. t G2 GE The behavior of the D2-minimum-distance as a function of t is quite different, as we now see. Lemma4.AcodeC fromConstruction1hasD -minimum- k−2t GI 3 2 distanced =2(n−k−t+1). G = 2 3 t GE G1 The proof of Lemma 4 is trivial from the fact that δ = t GE G2 n−k−t+1. Unfortunately,the construction does not exclude codewords with two minimum-weight sub-block codewords. Figure1. Generatormatricesforah3n,3k,3imulti-blockinterleaved The problem lies in the fact that both v and v are code C . 1,1 2,2 3 multiplied by the same matrixGE in the right column,which First note that the dimensions of the codes specified for may cancel their contribution to the right sub-block. We now G1,G2,GI,GE fit their correspondingsub-matrices in Fig. 1. summarize the distance properties in the following theorem It is straightforward to write a generator matrix for any code (proof immediate). of the type RS([a : a+ℓ −1] ), because the codewords of n this code are evaluations of polynomials of the form xaU(x), 1The bound is given on the minimum distance, but from the proof of whereU(x)isapolynomialofdegreeℓ−1orless.Asrequired, Theorem1itreadily applies tothe D1-minimum-distance aswell. Theorem5. A code C fromConstruction1 hassub-block Construction1)mapsk+tinformationsymbolstoeachoftwo 3 minimumdistanceδ = n − k − t + 1,D -minimum-distance sub-blocks, and k−2t to the third. 1 d =n−k+2t+1,andD -minimum-distanced =2(n−k−t+1). Fo1rt6(n−k+1)/4,C h2asminimumdistance2d =d (optimal), IV. AnImprovedConstructionwithLargerMinimum 3 1 Distance andfor(n−k+1)/4 < tithasminimumdistanced = d (not 2 optimal). To get codes hN,K,mi with larger minimum distances beyond what Construction 1 can achieve, we present the Similar to the case here, all the known I-I constructions following improved construction. In the sequel we define in the literature with optimal D -minimum-distance (given 1 s=t/2, assuming even t. t) suffer from the same problem of having codewords with two minimum-weightsub-blockcodewords.In[5]the authors Construction2.LetK bedefinedbyageneratormatrixG = 3 address this exact problem,but give no explicit constructions.  G  1 Tdheanimdp−li2ctatiinondo,fiTshtehoartemthe5re, iins paarrtaitciuolaorfth1ebteetrwmesen2tthine  GG2 ,whereG1,G2 andG3 arek × 3n matricesspecified 1 2 3 loss in d to the gain in d as we increase t. This high ratio in Fig. 2 as follows. The figure specifies 4s rows of each 2 1 compromises the code’s ability to address high error/erasure Gi,whicharepopulatedbythegeneratormatricesspecified eventsinbothoneandtwosub-blocks.Inournextconstruction below,orcombinationsthereof.Thefigureomitsthetopk−4s wesolvethisbyprovingamuchlowerratioof2/3.Butbefore rowsofeachGi thathosttheGI matrixgivenbelow,inthe that, we want to show the good property of Construction 1 correspondingsub-block2. having an extremely efficient reverse mapping from the sub- • G1isageneratormatrixforthecodeRS([0: s−1]n). block codeword to the k information symbols of the input • G2isageneratormatrixforthecodeRS([s:2s−1]n). sub-unit vj. We now write the encoding and reverse-mapping • G3isageneratormatrixforthecodeRS([2s:3s−1]n). functions explicitly. • G4isageneratormatrixforthecodeRS([3s:4s−1]n). Encoding: The encoder input is an information vector v = • GIisageneratormatrixforthecodeRS([4s:k−1]n). (v1,v2,v3), where each vj is a vector of k GF(q) symbols • GEisageneratormatrixforthecodeRS([k:k+s−1]n). holding the information sub-unit for sub-block j. The output • GFisageneratormatrixforthecodeRS([k+s:k+2s− of the encoderis simply the vector v·G, whichis a codeword 1] ). n ofC .ThematricesG1,G2,GI,GEarechosenasthestandard 3 polynomial-evaluation matrices of the respective RS codes n n n (mappingℓ polynomialcoefficients to n code symbols). For a v1,1 : G1 GF code RS([a : a+ℓ−1]n, the corresponding generator matrix v1,2 : G2 GE 4s mapsaninformationsub-vectorutoacodewordbyevaluating v1,3 : G3 GE thepolynomialU˜(x)= xaU(x)onn elementsofGF(q),where v1,4 : G4+G3 GE U(x) is the polynomialwhose coefficientsare the elements of v2,1 : G1 GF u, and it has degree at most ℓ−1. Note that this is not the v2,2 : G2 GE 4s way the RS code was defined in Definition 3, but it is true by v2,3 : GE G3 an equivalent definition of RS codes. v2,4 : GE G4+G3 Reversemapping:Thereverse-mappingoperationreturnsthe v3,1 : GF G1 information sub-unit vj from the codeword of sub-block j. v3,2 : GE G2 4s Denote by u(x) the polynomial whose coefficients are the v3,3 : GE G3 codeword symbols corresponding to u. From the property v3,4 : GE G4+G3 given in Definition 3, u(α−i) is zero for all i except those in left center right [a:a+ℓ−1] . Since these sets are not overlapping between n Figure2. Generatormatricesforanimprovedh3n,3k,3imulti-block G1,G2,GI,GE, we can take the combined sub-block code- interleaved code K . 3 wordpolynomialc(x)(whichisasumofdifferentpolynomials like u(x) above), and get that for each i∈[0 : k − 1], that Asinthepreviousconstructionthedimensionsofthecodes c(α−i) = u(α−i) for one of the sub-vectors u in v . Now we specified for G1,G2,G3,G4,GI,GE,GF fit their correspond- j can use the well known property of RS codes (inverse DFT) ing sub-matrices in Fig. 2. The code dimension of K3 is 3k, and obtain and its length is 3n. We prove the minimum distances in the following. U =n−1u(α−(a+l))=n−1c(α−(a+l)), l∈[0:ℓ−1]. l Proposition6. When t < k/2, K hassub-blockminimum 3 Since c(x) is given, we can find the coefficients of all the distanceδ=n−k−t+1. polynomials U(x) and thus all the vectors u. Proof: The proof is identical to Proposition 2: at each sub- Neither the I-I codes from [6] nor the ones in [5] have blockthereisacodewordofanRScodewithn−k−2s=n−k−t a known way to map k information symbols to each sub- roots with consecutive powers. block codeword, such that it is possible to retrieve these k symbols back from the sub-block codeword. The encoder specified forI-Icodes[1] (correspondingto the parametersof 2sameasinFig.1. Lemma7.ThecodeK hasD -minimum-distanced =n−k+ properties by lower bounding the minimum distances. How- 3 1 1 3t/2+1. ever, an alternative way is by giving an explicit erasure- decodingalgorithm,whichwedonow.First,supposethatone Proof:Recallthe partitionv=(v ,v ,v ).Nowwe partition 1 2 3 sub-blockhasn−k+3t/2erasuresandtheothertwohaveeach each v to j n−k−t erasures or less. From symmetry assume that the left vj =(vj,I,vj,1,vj,2,vj,3,vj,4). sub-block has the largest number. Then decoding the center sub-blockwitha[n,k+t]codegivesv ,...,v ,andsimilarly 2,1 2,4 The lengths of the constituent sub-vectors are from left to the right block gives v ,...,v . In addition, we get v as 3,1 3,4 1,1 right: k −4s,s,s,s,s. Consider a codeword of K with one 3 the reverse map of GF in the center sub-block, and then v 1,2 non-zero sub-block, for which we seek a lower bound on the by canceling v and v from the reverse map of GE in the 3,3 3,4 Hamming weight. Assume (wlog due to symmetry) that the center sub-block. Finally, we can find v +v by canceling 1,3 1,4 one non-zero sub-block is the left one. Then it is clear from v from the reverse map of GE in the right sub-block. Now 2,2 Fig. 2 that the only sub-vectors that can be non-zero are v , 1,I we can decode the left sub-block with a [n,k − 3t/2] code v ,andv . Settinganyothersub-vectorto non-zeroimplies 1,3 1,4 having contributions from only GI and G4. Second, suppose a non-zero codeword in at least one other sub-block. Further, thattwosub-blockshaveeachn−k−t/2erasuresandthethird ifv orv arenon-zero,thentheymustsatisfyv +v =0. 1,3 1,4 1,3 1,4 one hasn−k−t erasuresor less. From symmetryassume that Otherwisethecodewordintherightsub-blockwillbenon-zero the right sub-block has the smaller number. Then decoding aswell.Itcanbeseenthatthislastconditionimpliesthatonly the right sub-block with a [n,k+t] code gives v , and v 2,1 3,1 G4 contributes to the non-zero codeword, and together with (amongallv ).Nowaftercancelingthesetwowecandecode 3,l the contributionofGI fromv , we getn−k+3s consecutive 1,I each of the left and center sub-blockswith a [n,k+t/2] code roots and d =n−k+3t/2+1. 1 and recover their erased symbols. Lemma8.ThecodeK3hasD2-minimum-distanced2 = 2(n− V. Conclusion k−t/2+1). The constructions in this paper are the first known ones Proof: We use the definition of v from the proof of to offer read access in both sub-blocks and full blocks. The Lemma 7. Consider a codeword of K3 with two non-zero first construction has optimal D1-minimum-distance given t, sub-blocks, for which we seek a lower bound on the total but a limiting reduction in the D2-minimum-distance as t Hamming weight. Assume (wlog due to symmetry) that the grows. The second construction has slower growth of the D1- two non-zero sub-blocks are the left and center ones. Then minimum-distance, but in return also slower reduction in D2, it is clear from Fig. 2 that the only sub-vectors that can be and altogether a higher maximal minimum distance. Future non-zeroarev ,v ,v ,v ,v fromv ,andv ,v ,v , work is needed to extend these constructions, and find others 1,I 1,1 1,2 1,3 1,4 1 2,I 2,2 2,3 v from v . Other non-zero sub-vectors would imply a non- withsimilar tradeoffs,toadditionalmvalues.While thecodes 2,4 2 zero codeword on the right sub-block. The above restriction suffice to use finite fields with sizes as small as the sub-block on the sub-vectors that can be non-zero implies that there is size, further reducing the field size is an important future no contribution of F in the left sub-block codeword and no direction, e.g. through using BCH codes as the constituents. contributionofG1 in the center sub-blockcodeword.So each VI. Acknowledgement of the sub-block codewords has n−k− s consecutive-power roots, and thus d =2(n−k−s+1). This work was supported in part by the German-Israel 2 Foundation and by the Israel Science Foundation. CombiningProposition6 andLemmas7,8, we summarizethe distancepropertiesofConstruction2inthefollowingtheorem References (proof omitted). [1] M. Blaum and S. Hetzler, “Integrated interleaved codes as locally Theorem9. Acode K fromConstruction2hassub-block recoverable codes: properties and performance,” Int. J. Inf. Coding 3 Theory,vol.3,no.4,pp.324–344,January2016. minimumdistanceδ=n−k−t+1,D1-minimum-distanced1 = [2] M.Bossert,ChannelCodingforTelecommunications. Chichester,UK: n−k+3t/2+1,andD -minimum-distanced =2(n−k−t/2+1). Wiley,1999. 2 2 Fort 6 2(n−k+1)/5,K hasminimumdistanced = d ,and [3] U. Dettmar and U. Sorger, “New optimal partial unit memory codes 3 1 basedonextendedBCHcodes,”Electronicsletters,vol.29,no.23,pp. for2(n−k+1)/5<t 6(n−k+1)/2ithasminimumdistance 2024–2025, November1993. d =d . [4] P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, “On the locality 2 of codeword symbols,” IEEE Trans. Inf. Theory, vol. 58, no. 11, pp. Theorem9demonstratesthevalueofConstruction2:itallows 6925–6934, November2012. [5] J. Han and L. Lastras-Montano, “Reliable memories with subline ac- to use large values of t that give minimum distances higher cesses,”inIEEEInternationalSymposiumonInformationTheory(ISIT), than Construction 1 can reach. It can be calculated that when Nice,France, June2007. t > 2(n − k + 1)/7, Construction 2 has superior minimum [6] M. Hassner, K. Abdel-Ghaffar, A. Patel, R. Koetter, and B. Trager, “Integrated interleaving – a novel ECC architecture,” IEEE Trans. distance compared to Construction 1. From t=2(n−k+1)/7 Magn.,vol.37,no.3,pp.773–775,February2001. we can increase t in Construction 2 up to t = 2(n−k+1)/5, [7] G.Kamath,N.Prakash,V.Lalitha,andP.V.Kumar,“Codeswithlocal and getminimumdistance ashigh as d =1.6(n−k+1),while regeneration anderasure correction,” IEEETrans.Inf. Theory, vol. 60, no.8,pp.4637–4660,August2014. Construction 1 has a cutoff at d =1.5(n−k+1). [8] G. Lauer, “Some optimal partial-unit-memory codes (corresp.),” IEEE Erasure decoding: We chose to prove the code correction Trans.Inf.Theory,vol.25,no.2,pp.240–243,March1979. [9] L.-n.Lee,“Shortunit-memorybyte-orientedbinaryconvolutionalcodes having maximal free distance (corresp.),” IEEE Trans. Inf. Theory, vol.22,no.3,pp.349–352, May1976. [10] I.TamoandA.Barg, “Afamilyofoptimal locally recoverable codes,” IEEETrans.Inf.Theory,vol.60,no.8,pp.4661–4676, August2014.