Recall: ECC Approach: Redundancy. General Idea: Code Vector Space. Some Code Types Linear Codes: CS252 Graduate Computer Architecture Lecture 24

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1 Graduate omputer Archtecture Lecture Error orrecto odes Aprl rd, Joh Kubatowcz Electrcal Egeerg ad omputer ceces Uversty of alfora, Bereley Recall: E Approach: Redudacy Approach: Redudacy Add extra formato so that we ca recover from errors a we do better tha just create complete copes? Bloc odes: Data oded blocs data bts coded to ecoded bts Measure of overhead: Rate of ode: K/N Ofte called a (,) code osder data as vectors GF() [.e. vectors of bts ] ode pace s set of all vectors, Data space set of vectors Ecodg fucto: =f(d) Decodg fucto: d=f( ) Not all possble code vectors,, are vald! // cs-, Lecture Geeral Idea: ode Vector pace ode pace ode Dstace (Hammg Dstace) =f(v ) v Not every vector the code space s vald Hammg Dstace (d): Mmum umber of bt flps to tur oe code word to aother Number of errors that we ca detect: (d-) Number of errors that we ca fx: ½(d-) // cs-, Lecture ome ode Types Lear odes: G d H ode s geerated by G ad ull-space of H (,) code: Data space, ode space (,,d) code: specfy dstace d as well Radom code: Need to both detfy errors ad correct them Dstace d correct ½(d-) errors Erasure code: a correct errors f we ow whch bts/symbols are bad Example: RAID codes, where symbols are blocs of ds Dstace d correct (d-) errors Error detecto code: Dstace d detect (d-) errors Hammg odes d = olums ozero, Dstct d = olums ozero, Dstct, Odd-weght Bary Golay code: based o quadratc resdues mod Bary code: [,, 8] ad [,, ]. Ofte used space-based schemes, ca correct errors // cs-, Lecture

2 Hammg Boud, symbols GF() osder a (,) code wth dstace d How do,, ad d relate to oe aother? Frst questo: How bg are spheres? For dstace d, spheres are of radus ½ (d-),».e. all error wth weght ½ (d-) or less must ft wth sphere Thus, sze of sphere s at least: + Num(-bt err) + Num(-bt err) + + Num( ½(d-) bt err) ze ( d ) e e Hammg boud reflects b-pacg of spheres: eed of these spheres wth code space ( d ) e e ( ), d // cs-, Lecture How to Geerate code words? osder a lear code. Need a Geerator Matrx. Let v be the data value ( bts), be resultg code ( bts): Are there uque code values? Oly f the colums of G are learly depedet! Of course, eed some way of decodg as well. v G v f d ' G must be a matrx Is ths lear??? Why or why ot? A code s systematc f the data s drectly ecoded wth the code words. Meas Geerator has form: I a always tur o-systematc G code to a systematc oe (row ops) P But What s dstace of code? Not Obvous! // cs-, Lecture Implctly Defg odes by hec Matrx osder a party-chec matrx H ([-]) Defe vald code words as those that gve = (ull space of H) H ze of ull space? (ull-ra H)= f (-) learly depedet colums H uppose we trasmt code word wth error: Model ths as vector E whch flps selected bts of to get R (receved): R E osder what happes whe we multply by H: H R H ( E) H E What s dstace of code? ode has dstace d f o sum of d- or less colums yelds I.e. No error vectors, E, of weght < d have zero sydromes o ode desg s desgg H matrx // cs-, Lecture How to relate G ad H (Bary odes) Defg H maes t easy to uderstad dstace of code, but hard to geerate code (H defes code mplctly!) However, let H be of followg form: P s (-), I s (-)(-) H P I Result: H s (-) The, G ca be of followg form (maxmal code sze): I G P P s (-), I s Result: G s Notce: G geerates values ull-space of H ad has depedet colums so geerates uque values: H I G v P I v P // cs-, Lecture 8

3 // 9 cs-, Lecture mple example (Party, d=) Party code (8-bts): Note: omplexty of logc depeds o umber of s row! H G v v v v v v v v + c 8 + s 8 // cs-, Lecture mple example: Repetto (votg, d=) Repetto code (-bt): Postves: smple Negatves: Expesve: oly % of code word s data Not paced Hammg-boud sese (oly D=). ould get much more effcet codg by ecodg multple bts at a tme H G Error v // cs-, Lecture Bary Hammg code meets Hammg boud Recall boud for d=: o, rearragg: Thus, for: c= chec bts, (Repetto code) c= chec bts, c= chec bts,, use =8? c= chec bts,, use =? c= chec bts,, use =? c= chec bts,, use =? H matrx cossts of all uque, o-zero vectors There are c - vectors, c used for party, so remag c -c- Example: Hammg ode (d=) H G ) ( c c c ), ( // cs-, Lecture Example, d= code (E-DED) Desg H wth: All colums o-zero, odd-weght, dstct» Note that odd-weght refers to Hammg Weght,.e. umber of zeros Why does ths geerate d=? Ay sgle bt error wll geerate a dstct, o-zero value Ay double error wll geerate a dstct, o-zero value» Why? Add together two dstct colums, get dstct result Ay trple error wll geerate a o-zero value» Why? Add together three odd-weght values, get a odd-weght value o: eed four errors before dstgushable from code word Because d=: a correct error (gle Error orrecto,.e. E) a detect errors (Double Error Detecto,.e. DED) Example: Note: log sze of ullspace wll be (colums ra) =, so:» Ra =, sce rows depedet, cols dpt» learly, 8 bts code word» Thus: (8,) code

4 Twees: No reaso caot mae code shorter tha requred uppose -=8 bts of party. What s max code sze () for d=? Maxmum umber of uque, odd-weght colums: = 8 o, = 8. But, the = ( ) =. Werd! Just throw out colums of hgh weght ad mae (, ) code! rcut optmzato: f throwg out colum vectors, pc oes of hghest weght (# bts=) to smplfy crcut But shorteed codes le ths mght have d > some specal drectos Example: Kaeda paper, catches falures of groups of bts Good for catchg chp falures whe DRAM has groups of bts What about EVENODD code? a be used to hadle two erasures What about two dead DRAMs? Yes, f you ca really ow they are dead // cs-, Lecture Admstrva Mdterm Results: Almost doe. Really! Oe last problem to grade DIVA: A Relable ubstrate for Deep ubmcro Mcroarchtecture Desg, Author: Todd M. Aust Use of hecer stage placed after prmary computatoal stage Geeral addto of dyamc checg to OOO ppele Traset Fault Detecto va multaeous Multthreadg, Authors: teve K. Rehardt ad ubhedu. Muherjee Pared threads duplcatg computato to catch traset errors // cs-, Lecture How to correct errors? osder a party-chec matrx H ([-]) ompute the followg sydrome gve code elemet : H H E uppose that two correctable error vectors E ad E produce same sydrome: H E E H E E H But, sce both E ad E have (d-)/ bts set, E + E d- bts set so ths cocluso caot be true! o, sydrome s uque dcator of correctable error vectors E has d or more bts set E // cs-, Lecture // cs-, Lecture

5 Galos Feld Defto: Feld: a complete group of elemets wth: Addto, subtracto, multplcato, dvso ompletely closed uder these operatos Every elemet has a addtve verse Every elemet except zero has a multplcatve verse Examples: Real umbers Bary, called GF() Galos Feld wth base» Values,. Addto/subtracto: use xor. Multplcatve verse of s Prme feld, GF(p) Galos Feld wth base p» Values p-» Addto/subtracto/multplcato: modulo p» Multplcatve Iverse: every value except has verse» Example: GF(): mod, mod, mod Geeral Galos Feld: GF(p m ) base p (prme!), dmeso m» Values are vectors of elemets of GF(p) of dmeso m» Add/subtract: vector addto/subtracto» Multply/dvde: more complex» Just le real umbers but fte!» ommo for computer algorthms: GF( m ) // cs-, Lecture pecfc Example: Galos Felds GF( ) osder polyomals whose coeffcets come from GF(). Each term of the form x s ether preset or abset. Examples:,, x, x, ad x + x + = x + x + x + x + x + x + x + x Wth addto ad multplcato these form a rg (ot qute a feld stll mssg dvso): Add : XOR each elemet dvdually wth o carry: x + x + + x + + x + + x + x x + x + Multply : multplyg by x s le shftg to the left. x + x + x + x + x + x + x + x x + // cs-, Lecture 8 o what about dvso (mod) x + x x x + x + X + = x + x wth remader = x + x wth remader X + x + x + x + x + x + x Remader // cs-, Lecture 9 x + x x + x x + x x + x + x + x + Producg Galos Felds These polyomals form a Galos (fte) feld f we tae the results of ths multplcato modulo a prme polyomal p(x) A prme polyomal caot be wrtte as product of two o-trval polyomals q(x)r(x) For ay degree, there exsts at least oe prme polyomal. Wth t we ca form GF( ) Every Galos feld has a prmtve elemet,, such that all o-zero elemets of the feld ca be expressed as a power of erta choces of p(x) mae the smple polyomal x the prmtve elemet. These polyomals are called prmtve For example, x + x + s prmtve. o = x s a prmtve elemet ad successve powers of wll geerate all o-zero elemets of GF(). Example o ext slde. // cs-, Lecture

6 Galos Felds wth prmtve x + x + = = x = x = x = x + = x + x = x + x = x + x + 8 = x + 9 = x + x = x + x + = x + x + x = x + x + x + = x + x + = x + = Prmtve elemet α = x GF( ) α = x mod x + x + = x xor x + x + = x + I geeral fdg prmtve polyomals s dffcult. Most people just loo them up a table, such as: // cs-, Lecture Prmtve Polyomals x + x + x + x + x + x + x + x + x + x + x + x + x 8 + x + x + x + x 9 + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x 8 + x + x 9 + x + x + x+ x + x + x + x + Hardware shft left x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x 8 + x + x 9 + x + x + x + x + x + x + x + x + x + x + x + Galos Feld Multplcato by x Tag the result mod p(x) XOR-g wth the coeffcets of p(x) whe the most sgfcat coeffcet s. Obtag all - o-zero elemets by evaluatg x for =,, - hftg ad XOR-g - tmes. // cs-, Lecture Reed-olomo odes Galos feld codes: code words cosst of symbols Rather tha bts Reed-olomo codes: Based o polyomals GF( ) (I.e. -bt symbols) Data as coeffcets, code space as values of polyomal: P(x)=a +a x + a - x - oded: P(),P(),P().,P(-) a recover polyomal as log as get ay of Propertes: ca choose umber of chec symbols Reed-olomo codes are maxmum dstace separable (MD) a add d symbols for dstace d+ code Ofte used erasure code mode: as log as o more tha - coded symbols erased, ca recover data de ote: Multplcato by costat GF( ) ca be represeted by matrx: ax Decompose uow vector to bts: x=x +x x - Each colum s result of multplyg a by // cs-, Lecture Reed-olomo odes (co t) Reed-solomo codes (No-systematc): Data as coeffcets, code space as values of polyomal: P(x)=a +a x + a x oded: P(),P(),P().,P() alled Vadermode Matrx: maxmum ra Dfferet represetato (Ths H ad G ot related) lear that all combatos of two or less colums depedet d= Very easy to pc whatever d you happe to wat: add more rows Fast, ystematc verso of Reed-olomo: auchy Reed-olomo, others G ' H // cs-, Lecture a a a a a

7 Asde: Why erasure codg? Hgh Durablty/overhead rato! tatstcal Advatage of Fragmets Tme to oalesce vs. Fragmets Requested (TI) 8 Fracto Blocs Lost Per Year (FBLPY) Latecy 8 Explot law of large umbers for durablty! moth repar, FBLPY: Replcato:. Fragmetato: - // cs-, Lecture Objects Requested Latecy ad stadard devato reduced: Memory-less latecy model Rate ½ code wth total fragmets // cs-, Lecture ocluso E: add redudacy to correct for errors (,,d) code bts, data bts, dstace d Lear codes: code vectors computed by lear trasformato Erasure code: after detfyg erasures, ca correct Reed-olomo codes Based o GF(p ), ofte GF( ) Easy to get dstace d+ code wth d extra symbols Ofte used erasure mode // cs-, Lecture