Multi-Block Interleaved Codes for Local and Global Read Access

0.09 MB·

by  Yuval Cassuto

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).
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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.
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