Cooling Codes: Thermal-Management Coding for~High-Performance Interconnects

0.59 MB·

by  Yeow Meng Chee

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Cooling Codes: Thermal-Management Coding
for High-Performance Interconnects
Yeow Meng Cheex, Tuvi Etzion∗, Han Mao Kiahx, Alexander Vardy+x
xSchool of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
∗Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel,
the work was done while the author was visiting SPMS, Nanyang Technological University, Singapore,
and the Department of Electrical and Computer Engineering, University of California
7 +Department of Electrical and Computer Engineering, University of California, San Diego, CA 92093, USA
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0 [e-mail], [e-mail], [e-mail], [e-mail]
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n
a
J

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Abstract
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T
I High temperatures have dramatic negative effects on interconnect performance and, hence, numerous techniques
.
s havebeenproposedtoreducethepowerconsumptionofon-chipbuses.However,existingmethodsfallshortoffully
c
[ addressingthethermalchallengesposedbyhigh-performanceinterconnects.Inthispaper,weintroducenewefficient

1 coding schemes that make it possible to directly control the peak temperature of a bus by effectively cooling its
v
hottest wires. This is achieved by avoiding state transitions on the hottest wires for as long as necessary until their
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7 temperature drops off. At the same time, we reduce the average power consumption by making sure that the total
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numberofstatetransitionsonallthewiresisbelowaprescribedthreshold.Thesetwofeaturesareobtainedseparately
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0 or simultaneously. In addition, error-correction for the transmitted information can also be provided with each one
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1 of the two features and when they both obtained simultaneously.
0 Oursolutionscallforsomeredundancy:weuse n>k wirestoencodeagiven k-bitbus.Therefore,itisimpor-
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1 tanttodeterminetheoreticallytheminimumpossiblenumberofwiresnneededtoencodek bitswhilesatisfyingthe
:
v desired properties. We provide full theoretical analysis in each case, and show that the number of additional wires
Xi required to cool the t hottest wires becomes negligible when k is large. Moreover, although the proposed thermal-
r management techniques make use of sophisticated tools from combinatorics, discrete geometry, linear algebra, and
a
coding theory, the resulting encoders and decoders are fully practical. They do not require significant computational
overhead and can be im- plemented without sacrificing a large circuit area.
I. INTRODUCTION
Powerandheatdissipationlimitshaveemergedasafirst-orderdesignconstraintforchips,whethertargetedforbat-
tery-powereddevicesorforhigh-endsystems.Withthemigrationtoprocessgeometriesof65nmandbelow,power
dissipation has become as important an issue as timing and signal integrity. Aggressive technology scaling results
in smaller feature size, greater packing density, increasing microarchitectural complexity, and higher clock frequen-
cies. This is pushing chip level power consumption to the edge. It is not uncommon for on-chip hot spots to have
temperatures exceeding 100°C, while inter-chip temperature differentials often exceed 20°C.
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Power-aware design alone is not sufficient to address this thermal challenge, since it does not directly target
the spatial and temporal behavior of the operating environment. For this reason, thermally-aware approaches have
emerged as one of the most important domains of research in chip design today.
High temperatures have dramatic negative effects on circuit behavior, with interconnects being among the most
impacted circuit components. This is due, in part, to the ever decreasing interconnect pitch and the introduction
of low-k dielectric insulation which has low thermal conductivity. For example, as shown in [3], the Elmore delay
of an interconnect increases 5% to 6% for every 10°C increase in temperature, whereas the leakage current grows
exponentiallywithtemperature.Therefore,minimizingthetemperatureofinterconnectsisofparamountimportance
for thermally-aware design.
Figure1. Blockdiagramoftheproposedbusarchitecture
A. Related Work
Numerous encoding techniques have been proposed in the literature [4,9,10,28,35,39,41,46,47] in order to reduce
the overall power consumption of both on-chip and off-chip buses. It is well established [11,31,40,42,46,47] that
bus power is directly proportional to the product of line capacitance and the average number of state transitions
on the bus wires. Thus the general idea is to encode the data transmitted over the bus so as to reduce the aver-
age number of transitions. For example, the “bus-invert” code of [41] potentially complements the data on all the
wires, according to the Hamming distance between consecutive transmissions, thereby ensuring that the total num-
berofstatetransitionsonnbuswiresneverexceedsn/2.Unfortunately,encodingtechniquesdesignedtominimize
power consumption, do not directly address peak temperature minimization. In order to reduce the temperature of
a wire, it is not sufficient to minimize its average switching activity. Rather, it is necessary to control the temporal
distribution of the state transitions on the wire. To reduce the peak temperature of an interconnect, it is necessary
to exercise such control for all of its constituent wires.
In [46,47], the authors propose a thermal spreading approach. They present an efficient encoding scheme that
evenly spreads the switching activity among all the bus wires, using a simple architecture consisting of a shift-
register and a crossbar logic. This is designed to avoid the situation where a few wires get hot while the majority
are at a lower temperature. This spreading approach is further extended in [9,39] using on-line monitoring of the
switching activity on all the wires. Thermal spreading can be regarded as an attempt to control peak temperature
indirectly, by equalizing the distribution of signal transitions over all the wires.
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B. Our Contributions
Astechnologycontinuestoscale,existingmethodsmayfallshortoffullyaddressingthethermalchallengesposedby
high-performance interconnects in deep submicron (DSM) circuits. In this paper, we introduce new efficient coding
schemesthatsimultaneouslycontrolboththepeaktemperatureandtheaveragepowerconsumptionofinterconnects.
The proposed coding schemes are distinguished from existing state-of-the-art by having some or all of the fol-
lowing features:
A. Wedirectlycontrolthepeaktemperatureofabusbyeffectivelycoolingitshottestwires.Thisisachievedby
avoidingstatetransitionsonthehottestwiresforaslongasnecessaryuntiltheirtemperaturedecreases.
B. Wereducetheoverallpowerdissipationbyguaranteeingthatthetotalnumberoftransitionsonthebuswires
isbelowaspecifiedthresholdineverytransmission.
C. We combine properties A and/or B with coding for improved reliability (e.g., for low-swing signaling),
usingexistingerror-correctingcodes.
Toachievethesedesirablefeatures,weproposetoinsertattheinterfaceofthebusspecializedcircuitsimplementing
the encoding and decoding functions, denoted herein by E and D, respectively. This is illustrated in Figure1. The
variouscodingschemesintroducedinthispaperemploytoolsfromvariousfieldssuchascombinatorics,graphtheory,
block designs, discrete geometry, linear algebra, and the theory of error-correcting codes. Nonetheless, in each case
theresultingencoders/decodersE andD areefficient:theydonotrequiresignificantcomputationaloverheadandcan
beimplementedwithoutsacrificingalargecircuitarea.ThisisespeciallytrueforPropertyA,wherethecomplexityof
encodinganddecodingscaleslinearlywiththenumberofwires.
Weconsiderbothadaptiveandnonadaptive(memoryless)codingschemes.Theadvantageofnonadaptiveschemes
is thatthey are easierto implement anddo not requirememory. The disadvantageis thatit is notpossible to imple-
mentPropertyAwithnonadaptiveencoding.Forthisreason,mostofthecodingschemesdevelopedinthispaperwill
beadaptive,basedontheideaof differential encoding.Notably,however,allofourschemesrequiretheencoderand
decodercircuitstokeeptrackof only one(themostrecent)previoustransmission.
Unlike the thermal spreading methods of [39,46,47] that lead to irredundant coding schemes, the solutions we
proposedointroduceredundancy:werequiren > kwirestoencodeagivenk-bitbus.Akeyconsiderationinthissit-
uationisthe area overhead due to the additional n−k wires.Therefore,itisimportanttodeterminethetheoretically
minimumpossiblenumberofwiresnneededtoencodekbitswhilesatisfyingthedesiredproperties.Weprovidefull
theoreticalanalysisineachcase.WemoreovershowthatthenumberofadditionalwiresrequiredtosatisfyPropertyA
becomesnegligiblewhenkislarge.
C. Thermal Model
Chiang,Banerjee,andSaraswat[11]cameupwithananalyticmodelthatcharacterizesthermaleffectsduetoJoule
heatinginhigh-performanceCu/low-kinterconnects,underbothsteady-stateandtransientstressconditions.Shortly
thereafter,SotiriadisandChandrakasan[40]gaveapowerdissipationmodelforDSMbuses.Thesetwomodels,ac-
countingforthermalandpowereffectsseparately,werelaterunifiedandrefinedbySundaresanandMahapatra[42].
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Finally, building upon this work, Wang, Xie,Vijaykrishnan, and Irwin [47] proposed a more accurate thermal-and-
powermodelforDSMbuses.Inallthesepapers,ann-bitbus(illustratedinFigure2)ismodeledinterms
Rinter
…
wire wire wire
(1) (2) (n)
Rspread
tILD
heat flow to substrate layer Rrect
Figure2.GeometryusedforcalculatingRspread,Rrect,andRinter
Rinter
…
R1 C1 P1 R2 C2 P2 Rn Cn Pn
substrate at ambient temperature
Figure3.EquivalentthermalRC-networkforak-bitbus
oftheequivalentthermal-RCnetworkinFigure3.SundaresanandMahapatra[42]showthatthisthermal-RCnetwork
isgovernedbythefollowingdifferentialequations:
∂θ θ −θ θ −θ
P = C 1 + 1 0 + 1 2 , (1)
1 1
∂t R R
1 inter
P = C ∂θk + θk−θ0 + θk−θk−1 , and (2)
n k ∂t R R
k inter
P = C ∂θi + θi−θ0 + 2θi−θi−1−θi+1 (3)
i i
∂t R R
i inter
fori =2,3,...,n−1,whereP istheinstantaneouspowerdissipatedbywirei,C isthethermalcapacitanceperunit
i i
length of wire i, R = R +R is the thermal resistance per unit length of wire i along the heat transfer path
i spread rect
downwards,R isthelateralthermalresistanceusedtoaccountfortheparallelthermalcouplingeffectbetweenthe
inter
wires,θ isthetemperatureofwirei,andθ isthesubstrateambienttemperature.
i 0
Inanybusmodelforwhich(1)–(3)hold,thetemperatureofawirewillincreasewheneverthewireundergoesastatetran-
sition; conversely, in the absence of state transitions, the temperature will gradually decrease. We let σ denote the
i
switchingactivityofwirei,whichisthenumberoftimesthewirechangesstate.Thenthepowerdissipatedbyabus
isdeterminedbyits total switching activityσ +σ +···+σ .
1 2 n
In order to directly control the peak temperature of a bus by avoiding transitions on its hottest wires,we need to
knowwhichwiresarethehottestateverytransmission.Therearetwogeneralwaystoobtainthisinformation.Wecan
useananalyticalmodel[42],suchas(1)–(3),toestimatethecurrenttemperaturesofthewires.Foreachwire,such
anestimatecanbeimplementedwithacounterthatisincrementedontransitionanddecrementedonnon-transition,
where the precise magnitude of the increments/decrements is determined by the model. Alternatively, we can have
actualtemperaturesensorsforeachwire.ForDSMbuses,accuratetemperaturesensingcanbeimplementedusing,for
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example,ringoscillators[13].Asshownin[13],sensorsbasedonringoscillatorsprovidearesolutionof1°Cwhile
consuminganactivepowerofonly65–112µW.
D. Organization
Therestofthispaperisorganizedasfollows.Webegininthenextsectionwithapreciseformulationofthecoding
problems that result from the thermal-management features we propose to implement — namely, Properties A, B,
andC.InSectionIII,wepresentanonadaptivecodingschemethatcombinesPropertyB(reducingtheaveragepower
dissipation)withthethermalspreadingapproachof[47].OurconstructionsinSectionIIIarebasedonthenotionsof
anticodes and quorum systems, and use key results from the theory of combinatorial designs. SectionIV is devoted
to PropertyA: we show how state transitions on the t hottest wires can be avoided by using only t+1 additional
buslines.Thisoptimalconstructionisbasedoncombining differential codingwiththenotionof spreadsand partial
spreadsinprojectivegeometry.Theoptimalconstructioncanbeappliedwhent+1(cid:54)(n+1)/2.Whent+1> (n+
1)/2 we use another technique from the theory of error-correcting codes to construct efficient codes. The designed
codes can be viewed as sunflowers, while the partial spreads, are also sunflowers, and hence can be also viewed as
a special case of these codes. The technique used is generalized with the notion of generalized Hamming weights.
In SectionV, we show how Properties A and B can be all achieved at the same time. That is, we design coding
schemes that simultaneously control peak temperature and average power consumption in every transmission. For
this purpose, we present three types of constructions. The first construction is based upon the Baranyai theorem on
complete hypergraph decomposition into pairwise disjoint perfect matchings. The second construction is based on
concatenation of low weights codes based on appropriate non-binary dual codes or non-binary partial spreads. The
thirdconstructionistheprevioussunflowerconstruction,whichalsosatisfyPropertyB.SectionVIisdevotedtocodes
whichsatisfyPropertyCsimultaneouslywitheitherPropertyAorPropertyBorboth,i.e.weaddalsocorrectionfor
possible transmission errors on the bus wires. The constructions in this section will also be of two types. The first
type of constructions is based on resolutions in block design. The second type of constructions will be to employ
thepreviouslygivenconstructions,whereoursetoftransitionsisrestrictedtothesetofcodewordsinagivenerror-
correcting code. In all these sections our bounds and constructions are applied for infinite sets of parameters, but
thereisnoasymptoticanalysisinanyofthesecases.TheasymptoticanalysisispostponedtoSectionVII,wherethe
asymptoticbehaviorofourcodesisanalyzed.Inparticularweanalyzeareaoverheadofourconstructionsandprove
that when k is large enough, the additional number of wires required to satisfy the desired properties is negligible.
Finally, SectionVIII summarizes our comprehensive work and presents a brief discussion of possible directions for
futureresearch.
II. PROBLEMFORMULATIONANDPRELIMINARIES
Let us now elaborate upon Properties A, B, C introduced in the previous section. For each of these properties, we
willcharacterizetheperformanceofthecorrespondingcodingschemebya single integer parameter.Allofourcod-
ingschemeswillusen > kwirestoencodeak-bitbus.Weassumethatcommunicationacrossthebusissynchronous,
occurringinclockedcyclescalled transmissions.Thisleadstothefollowingdefinition.
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Definition1.Consideracodingschemeforcommunicationoverabusconsistingof n wires.Let t, w, e bepositive
integerslessthann.Wesaythatthecodingschemehas
PropertyA(t):ifeverytransmissiondoesnotcausestatetransitionsonthethottestwires;
PropertyB(w):ifthetotalnumberofstatetransitionsonallthewiresisatmostw,ineverytransmission;
PropertyC(e):ifuptoetransmissionerrors (0receivedas1,or 1receivedas0)onthenwirescanbecorrected.
Wepresumethat,atthetimeoftransmission,itisknownwhich t wiresarethehottest;PropertyA(t) isrequiredto
holdassumingthat anytwirescanbedesignatedasthehottest.
The values of t, w, e are design parameters, to be determined by the specific thermal requirements of specific
interconnects. The proposed coding schemes will work for various values of t, w, and e. Nevertheless, it might be
helpfultothinkoftasasmallconstant,sincesignificantreductionsinthepeaktemperaturecanbeachievedbycooling
onlyafewofthehottestwires.Thus,themostimportantvaluesoftaresmallones,say,lessthanhalfofthebuswires.
But,ourconstructionsinthefollowingsectionswillconsideralsosolutionsforlargevaluesoft,specifically,anyvalue
oft.Similarly,wisalsousuallysmallsincelargewmeansalargenumberofstatetransitionsonallthewires,which
mightresultintoomanyhotwires.Finally,wealsoexpectetobesmall,especiallyasitmustbesmallerthanw/2as
otherwisewewon’tbeabletocorrecttheerrors.
CodeswhichsatisfyPropertiesA(t),B(w),andC(e)simultaneouslyineverytransmission,willbecalled(n,t,w,e)-
low-powererror-correctingcoolingcodesor(n,t,w,e)-LPECCcodesforshort.Whenanonemptymeaningfulsubset
ofthethreeproperties(propertyC(e)isnotinterestingaloneinourcontext)willbesatisfied,onlytheparametersand
description related to this subset of properties will remain in the name. For example, (n,t,e)-LPEC codes stands
for (n,t,e)-low-power error-correcting codes. Six such nonempty subsets exist and for each one we suggest coding
schemes and related codes. It the conclusion of Section VIII a pointer will be given, where each one of these six
subsetswasconsidered.
Weviewthecollectivestateofthenwiresduringeachtransmissionasabinaryvectorx= (x ,x ,...,x ).Theset
1 2 n
ofallsuchbinaryvectorsisthe Hamming n-spaceH(n) = {0,1}n.WewillidentifyH(n)withthevectorspaceFn.
2
Givenanyx,y∈Fn,the Hamming distance d(x,y) isthenumberofpositionswherexandydiffer.The Hamming
2
weightofavectorx∈Fn,denotedwt(x),isthenumberofnonzeropositionsinx.
2
Conventionally,abinary code C of length n issimplyasubsetofFn.TheelementsofCarecalled codewords.
2
GivenacodeC,its minimum distanced(C)and diameterdiam(C)aredefinedasfollows:
d(C)d=ef min d(x,y) and diam(C)d=ef maxd(x,y).
x,y∈C x,y∈C
Later,inSectionsIVandV,wewillneedtomodifyandgeneralizethisconventionaldefinitionofbinarycodesinan
importantway.ThismodificationwillbeneededforcodeswhichsatisfyPropertyA(t).
III. NONADAPTIVELOW-POWERCODES
Theencodingschemesconsideredinthissectionbelongtothe nonadaptivekind,inthatthechoicewhichcodeword
totransmitacrossthebusinthecurrenttransmissiondoesnotdependoncodewordsthathavebeentransmittedearlier.
Suchcodingschemesarealsoknownas memoryless.Theadvantageofnonadaptiveschemesisthattheyaresimpler
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to implement: they do not need a continuously changing data model, and they do not require memory to track the
historyofprevioustransmissions.
Inthenonadaptivecase,an n-bit coding schemeforasourceS⊆Fk isatripleE= (cid:104)C,E,D(cid:105),where
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1) Cisabinarycodeoflengthn,
2) E:S→Cisabijectivemapcalledanencoding function,
3) D:C→Sisabijectivemapcalledadecoding function,suchthatD(cid:0)E(u)(cid:1) =uforallu∈S.
EncodinganddecodingcircuitsthatimplementE andDareinsertedattheinterfaceofthebus(seeFigure1).
Supposeu,v∈Saretwowordsthataretobecommunicatedacrossthebusduringconsecutivetransmissions.Inthe
absenceofacodingscheme,thetotalswitchingactivityofthebusisthengivenby|{i : u (cid:54)= v }|.Thisisprecisely
i i
the Hamming distance d(u,v), which could be as high as k. If an n-bit coding scheme is used, then x = E(u) and
y= E(v)aretransmittedinstead.Theresultingtotalswitchingactivityofthebusistherefored(x,y),whichisupper
boundedbydiam(C).
Itfollowsthatthecodingscheme satisfies PropertyB(w) if and only if diam(C)(cid:54)w.Asthepowerconsump-
tionofabusisdirectlyrelatedtoitstotalswitchingactivity,wecallsuchacodeCan(n,w)-low-power code((n,w)-
LPcodeforshort).
In this section, we are interested in (n,w)-LP codes that also achieve low peak temperatures by spreading the
switching activitiesamongthebuswiresasuniformlyaspossible.Indoingso,wearefollowingtheanalysisof[10,
46,47]andtheresulting thermal spreadingapproach[10,39,46].Inordertoquantifythethermalspreadingachieved
byagivencodingschemeE = (cid:104)C,E,D(cid:105),letustreatthesourceSasarandomvariabletakingonvaluesinFk,and
2
assumethatSisuniformlydistributed.Thisisacommonassumptioninbusanalysis—see,forexample,[40].With
thisassumption,theexpectedswitchingactivityofwireiisgivenby
µi = |S1|2 ∑ (cid:12)(cid:12)E(u)i−E(v)i(cid:12)(cid:12) = 2ri(||CS||2−ri) (4)
u,v∈S
where r is the number of codewords (x ,x ,...,x )∈C such that x = 1. If µ ,µ ,...,µ are all equal, we say
i 1 2 n i 1 2 n
that the code C is thermal-optimal, since the expected switching activities of the bus wires are then uniformly dis-
tributed.Thisleadstothefollowingproblem:
Givennandw,determinethemaximumsizeofathermal-optimal(n,w)-low-powercode (5)
The size of C is important because we wish to minimize the area overhead introduced by our coding scheme. This
overheadislargelydeterminedbythenumbern−kofadditionalwiresthatweneedtoencodeagivensourceS⊆Fk.
2
Clearly,toencodesuchasource,weneedacodeCwith|C|(cid:62)2k.
It is easy to see from (4) that µ ,µ ,...,µ are all equal if and only if r ,r ,...,r are all equal. Hence in a
1 2 n 1 2 n
thermal-opti- mal code C, the number of codewords (x ,x ,...,x )∈C having x =1 is the same for all i. Such
1 2 n i
codesaresaidtobeequireplicateinthecombinatoricsliterature.Toconstructsuchcodes,wewillneedtoolsfromthe
theoryofsetsystemsaswassuggestedbyChee,Colbourn,andLing[10].
A. Set Systems
Givenapositiveintegern,theset{1,2,...,n}isabbreviatedas[n].ForafinitesetXandk(cid:54)|X|,wedefine
(cid:18) (cid:19)
2X d=ef(cid:8)A : A ⊆ X(cid:9) and X d=ef(cid:8)A ∈2X : |A| = k(cid:9)
k
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A set system of order n is a pair (X,A), where X is a finite set of n points and A ⊆ 2X. The elements of A
arecalled blocks.Asetsystem(X,2X)isacomplete set system.The replication numberofx∈X isthenumberof
blockscontainingx.Asetsystemis equireplicateifitsreplicationnumbersareallequal.
There is a natural one-to-one correspondence between the Hamming n-space Fn and the complete set system
2
([n],2[n])ofordern.Foravectorx= (x ,x ,...,x )∈Fn,the support of xisdefinedas
1 2 n 2
supp(x) d=ef (cid:8)i ∈ [n] : x (cid:54)=0(cid:9)
i
With this, the positions of vectors in Fn correspond to points in [n], each vector x∈Fn corresponds to the block
2 2
supp(x), and d(x,y) = |supp(x)(cid:52) supp(y)|, where (cid:52) stands for the symmetric difference. It follows from the
above that there is a 1-1 correspondence between the set of all codes of length n and the set of all set systems of
ordern.Thuswemayspeakofthe set system of a codeorthe code of a set system.
B. Thermal-Optimal Low-Power Codes
Thesetsystem([n],A)ofathermal-optimal(n,w)-low-powercodeisdefinedbythefollowingproperties:
1) |A (cid:52)A |(cid:54)wforall A ,A ∈ A,and
1 2 1 2
2) ([n],A)isequireplicate.
Itfollowsthatourproblemin(5)canberecastasanequivalentprobleminextremalsetsystems,asfollows:
Givennandw,determineT(n,w),themaximumsizeofanequireplicateset
(6)
system(X,A)ofordernsuchthat|A (cid:52)A |(cid:54)wforall A ,A ∈A
1 2 1 2
Iftheequireplicationconditionisremoved,theresultingsetsystemisknownasan anticode of length n and diame-
ter w.Hence,thermal-optimallow-powercodesareequivalentto equi- replicate anticodes.Anticodesingeneral,and
thesizeofanticodesofmaximumsizehavebeenasubjectofintensiveresearchincodingtheory,see[1,2,8,15,32,
38]andreferencestherein.
Thedeterminationofequireplicateanticodesofmaximumsizeappearstobeanewproblem,alsotothecombina-
torics and coding theory communities. However, the maximum size of an anticode has been completely determined
byKleitman[27],andevenearlierbyKatona[25],inadifferentbutequivalentsetting.Thusthefollowingtheoremis
from[25]and[27].
Theorem1. Let T(n,w) be the maximum number of blocks in a set system ([n],A) with |A (cid:52)A | (cid:54) w for all
1 2
A1,A2∈A.Then  w∑/2(cid:18)ni(cid:19) if w ≡0 (mod 2)
T(n,w) = i=0
(cid:18)nw−−11(cid:19) + w∑−21(cid:18)ni(cid:19) if w ≡1 (mod 2)
2 i=0
Forallevenw,anextremalsetsystem([n],A)withT(n,w)blocksisgivenby:
w(cid:91)/2(cid:18)[n](cid:19)
A = (7)
i
i=0
Ifwisodd,letxbeanyfixedelementof[n].Thenanextremalsetsystem([n],A)isgivenby:
w(cid:91)−21(cid:18)[n](cid:19) (cid:91) (cid:40) (cid:18)[n]\{x}(cid:19)(cid:41)
A = A∪{x} : A ∈ (8)
i w−1
i=0 2
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Weobservehere thatwhen w iseven, theextremalsetsystem inTheorem1 isequireplicate. Itconsistsofallthe
vectorsoflengthnandweightatmostw/2.Hence,wehavethefollowingresult,whichsolves(5)and(6)forevenw.
Corollary2.
w∑/2(cid:18)n(cid:19)
T(n,w) = when w ≡0(mod 2)
i
i=0
The situation when w is odd is much more difficult. The set system in (8) is not equireplicate. In particular, we
do not know if there exists an equireplicate anticode of order n and diameter w having size T(n,w). Hence, from
Theorem1wecanderiveonlythatforalloddw,wehave:
w∑−21(cid:18)n(cid:19) (cid:54) T(n,w) (cid:54) (cid:18)n−1(cid:19) + w∑−21(cid:18)n(cid:19) (9)
i w−1 i
i=0 2 i=0
Thelefthandsideoftheequationisobtainedfromacodewhichconsistsofallthevectorsoflengthnandweightat
most(w−1)/2.TherighthandsideisobtainedfromtheupperboundonT(n,w)giveninTheorem1.
ThenextthreepropositionsestablishessomeexactvaluesofT(n,w)foroddw.
Proposition3.
T(n,1) = 1 for n(cid:62)2.
Proof:Sincethedistancebetweentwodifferentvectorsoflengthnandweightoneistwo,itfollowsthatT(n,1) =
1whenn(cid:62)2.Acodewithmaximumsizeconsistsoftheuniqueall-zerovectoroflengthn.
Proposition4.
T(n,n−1) = 2n−1 for n(cid:62)3 .
Proof: When the distance between two codeword is at most n−1, the code cannot contain two complement
codewordsandhenceitssizeisatmost2n−1.Foroddn,anequireplicatesetsystemofsize2n−1 isobtainedfromall
vectorsoflengthnandweightatmostn/2.Forevenn,wegivethefollowingconstruction(whichalsoworksforany
oddn(cid:62)5ifinductionisapplied).Let([n−1],A)beasetsystemwhichattainsT(n−1,n−2) =2n−2.Wedefine
thefollowingsetsystem([n],B).
B d=ef {X∪{n} : X ∈ A}(cid:91){X : X ∈ A}.
WeclaimthatB isanequireplicatesetsystemwhichattains T(n,n−1) = 2n−1.Clearly,|B| = 2|A| = 2n−1 and
the fact that A does not contain complement blocks immediately implies that also B does not contain complement
blocks. Finally, since A is equireplicate and its size is 2n−2, it follows that each element of [n−1] is contained in
2n−3 blocksofA.Now,iseasytoverifythatifeachi ∈ [n−1]iscontainedin2n−3 blocksofA,theneachi ∈ [n]
iscontainedin2n−2elementsinB.Hence,BisanequireplicatesetsystemwhichattainsT(n,n−1) =2n−1.
Proposition5.
T(n,3) = n+1 for n(cid:62)5.
Proof: Firstnote,thatbyProposition4wehaveT(4,3) = 8.Also,T(n,2) = ∑1 (n) = n+1foralln (cid:62) 2.
i=0 i
Finally, it is easy to verify that T(5,3) = 6, These facts will be used in the current proof that T(n,3) = n+1 if
n >4.LetCbethelargestpossibleequireplicateanticodeoflengthn >5anddiameter3.
10
LetxandzbetwocodewordsofCsuchthatd(x,z) = 3.W.l.o.g.xandzdifferinthelastthreecoordinatesand
thefirstn−3coordinatesinbothhavex1,x2,...,xn−3.
Letαβγandα¯β¯γ¯ bethevaluesofthelastthreecoordinatesinxandz,respectively.Thereisnoothercodewordin
Cwhichendswitheitherαβγorα¯β¯γ¯ sincesuchacodewordshouldalsostartwithx1,x2,...,xn−3,toavoiddistance
greaterthan3fromeitherxorz,andhencesuchacodewordwillbeequaltoeitherxorz.Sinceeachoneofthelast
threecolumnshasatleastone zeroandatleastone oneandtheanticodeisequireplicate,itfollowsthattheweightof
acolumnisatleast1andatmost |C|−1.TheHammingdistanceofanytwoof100,010,001,111is2andhence
theprefixesoflengthn−3relatedtothecodewordsendingwiththesesufficesdifferinatmostonecoordinate.Since
anticodewithdiameteronehasatmosttwocodewords,itfollowsthatallcodewordswiththesesuffices(ifdiffer)have
differentvaluesinexactlythesamecoordinate(intheprefixoflengthn−3).Thesameargumentholdsalsoforthe
codewordsendingwith011,101,110,and000(theyhavethesamevaluesinn−4outofthefirstn−3coordinates).
Notethatsincethesuffixofeitherxorzisin{100,010,001,111}andtheothersuffiexisin{011,101,110,000},it
followsthatalltheothern−5coordinates(whichdon’thavedifferentvalues)ofxandzhavethesamevaluesforall
thecodewords.Sincen > 5,itfollowsthateachoneofthen−5coordinates(whichexist)formseitheracolumnof
zeroesoracolumnof ones.Ifthecolumnconsistsof zeroeswehaveacontradictionsincetheweightofthecolumn
is0whichissmallerthan1.Ifthecolumnconsistsof oneswehaveacontradictionsincetheweightofthecolumnis
|C|whichisgreaterthan|C|−1.
Thus, for n > 5 there are no two codewords for which the Hamming distance is three. and hence for n > 6,
T(n,3) = T(n,2) = n+1,whichcompletestheproof.
For other odd values of w, we start with the extremal anticode A of diameter w−1 in (7) and add blocks to A
whilemaintainingtheequireplicationrequirement.Suchblocksmustcontainexactly (w+1)/2pointstomakesure
thattheirdistancewiththeblocksofAwith(w−1)/2points,orless,willnotexceedw+1.Anytwoblockswith
(w+1)/2 points must intersect in at least one point as otherwise their distance will be w+1. Interestingly, these
propertiespreciselydefinea regular uniform quorum systemofrank(w+1)/2.Quorumsystemshavebeenstudied
extensivelyintheliteratureonfault-tolerantanddistributedcomputing—see[45]forarecentsurvey.Therearemany
typesofsuchsystems.Forexample,if(w+1)/2 = q+1,n = q2+q+1,andqisaprimepower,thenanoptimal
suchsystemconsistsofq2+q+1blockswhichformaprojectiveplaneoforderq[44,p.224].Inotherwords
Proposition6.Ifw =2q+1,qisaprimepower,then
T(q2+q+1,2q+1) = ∑q (cid:18)q2+q+1(cid:19)+q2+q+1.
i
i=0
For a proof of this proposition and other similar constructions, we refer the reader to the work in [12], where such
systemsareconstructedfromcombinatorialdesigns.
IV. COOLINGCODES
Unfortunately, it is not possible to satisfy PropertyA(t) with nonadaptive coding schemes, even for t = 1. Indeed,
suppose we wish to avoid state transitions on just the one hottest wire, say wire i. If the encoder does not know
the current state of wire i, the only way to guarantee that there is no state transition is to have x = y for any
i i