BSTJ 40: 1. January 1961: Error-Correcting Codes for Multiple-Level Transmission. (MacWilliams, Jessie)

10.3 MB·English

Frror-Correcting Codes for
Multiple-Level Transmission

By JESSIE. MucW

A elovl alphas i jn a 0 ow aolr pace wer file fll wie
ardent The flere of alphabet ona of the sel parr, ok
ans of mba fr the grove od. The eight uf ete i
Inve of onsero some i amine. The mee weight of the ltrs
the alphabet, elt sri noted yd relonahip i exabihad

sien the elpholt ave a set of pots Sina nite projective apace. Phe
fea mena curosnctoner later the tere of the aihoet ad the
agerplaves of the pace. The weight of tee Sx apy reed te
fence oe set Sik the oreepontng harp

"Fo ats of pit in Se precive space ae call ewido iF hay
ave yeni of te spore. Ino ep a ald ep
ff then evita tena hom, a rose apace a ele presen =
Founacpdhi. Jt 30 shown Chat thee esnione mean the same thing nd
duce to hws dfaton ation 92

“nthe staid between the dinaion of the ape! and
the porated hie gi fr han fr mx reno the her
Dpevamelors,1¢ i town at thin baud ead be aired ly alkalis
teth erated eta A meth i gia fr eaarveing la of abe
each atin ths boc, 2 son Ha fr he ease = 2 tone
he only et {inthe see of pe) foe wich the awd bee

mt

1AM,

A gweat des! of work has Leen done on exror-onrecting sodes fr tha
biaty ohare. Tn This paper We censider codes fora ehunzal that ea
Ianeast mur than v0 levels, Mtplelevel trenton is practical
iT the carne! is suri quet, ae for example, the ubmaine voice
table I ml ia substantia eres Sn Ik re dina

ithity ning a cade. One nse hos four parameters toad — the
tuner uf Teele of seanerieion, the wumber of information esis,
"he br of dada yt, Ue ter of erry dee
able ta deiece and/or corect. OF couse i canuot be decided witha
dlevaled snalysis whether thece advancages il more than compenate
fer the ded romplesty of the terminal equipoen

To th binary tay, sylenie orroreuoreting codes he cet
esearch ne rene to knoe mathemati tes
aiques, Tt has been shown hy Sepia’ tht the words of « eyteratio
fork fun group sal plata ion ml ‘he nara
tononviation fa group nner the Fel (01) apne ta be a veer
cor ove faite Held of ¢ elena. We eal uch veer spaces aFpba-
Fete, and cher individual lemencs ae alld tues. Ia the general ue
cede bocomes an “pabet” wad a word (unfortunately!) Becton
t "letter." Rash iter it row of syle picker from the grourd
Fel; the altel iv see ro vectors of length w. The g eierent
symbole of the ground field correspond to 9 dffrest trmmamiscinn levels

‘eu only a neteetil typeof eke emilee, seme wenn
nt sale about the nar of the channel ard of the Fakormadion
ng (ranted. Those nea foi

{a} The numberof transaiesion level 2 poser of «prime mn,
tines the somber of eeraeata i «faite Eel a power nf rm, To
Dprantieg ths enol amvere rssicon; hotween one and nine we ve
‘elidel oly te auaber

{The vhante ie “synrmde” isthe see thas ever’symbel ee
‘he fame shance of gerting tvongh sorry, and that the proaliey
one syzbal bring hanged ini ane i the sti for every pa
sd

(Co) Allerome ae equally ad, This might be theese, for exemple, if
‘one mer ordering mershastise fra @ mailorder house by extalg mise
her only.

‘With te assumptions the pnwiples uf eer meetin Dye lel
Alphule ae maety shea ox Une deri Ty Sepirfoe rou
fea tie a to eve alphatet;- Fer eorwenience she peesinent celts
from Slew’ per ate ssatind in the Aggie, Phe purmnelers
fan spades, biden sn a

IIe dimengion asa vertor space, denoted by &. The alphabet con
‘eine lettre: palo the nursber of eymbosin eae eter whi ean
be reguced us eurrying information. The remaining n — F symbols axe
aldo forthe purpene of crrarleetion ana armen,

2. The iinimuin neh i the eters of Ue alphabet other tha
(0 --- 0). (The wail of «let the nantes of nonscro eyraboe it
tela.) The quantity ely edt the errors prop
inuneta-tavu uo conserva coms ose

crt ue alpbet; fan spats be capable of eomecting all
Srourmeeref 1,2. eetrnein ae Iter it must haved = 201

‘The parpove of thi pur so fvenigate the properties of weror
passe ower faite Sells, paral these proporties whith an related
Inthe parame:ce & The weigh getter ext oy Sa neation Yaa
particule suf The vitor saa, wich fxn weed situation in
denn alge, Henne nr chief rachenciel tole al sgh ba
Fite projective gcomelsy. The esnoston hotween inary goa wads
fn fice gore as pte out By Hue aa is many exended
to th xoseral cus

We fest ostublsh overs ew deft of oquvalonse between alpha
hts (To equivalene lphbes have the sane een-coreeting proper
thos) Tower bd for is Roe i tevin og dd. Ceely
Uleseuble to have w — (the sbmber of eheceymbais} as sal we
[eel Te is shown tha sis hoe bd ea Be attcne, bt aly By
2 steed else of pls. Ther alphabets arc, on the whole, not
Prarie: foe eommunention puepores unless the expecta cur ens
strmoely high Hewover, thr yeometis metheds used. the wnsteur
Fin shee alsbubet an espe to iad useful alphabets or specie
Saee "he chon dened for lvel alphabets mpply equally well 0
those g = 2am esrtafante wo the Mary a Wary geoup codes

In this scetion we deli He tation to be used in thie paper aaa
inneduce Boss's theorem en the roblion Lett alphabets and
projective gooneteien

Tet Pie int with g elements nnd charaeleite p, nd let
ey) denote the nonera elect Pq). We consider a vector space
fof dimension w over Bq). Tet (f(y) denote th “row space," tha
partiular representation of the veotor=pane comsstng of all pole

elopesa elowents of Pg), Ke example, Gl rnin of che 2rples
fom) cm Us ey Ce
0) Owed feed anh Go

fa) fe) FE Gok Sos
sve 8 primitive ene ral of cay

Clearly Gag has a° mecubers They” Lnensotoclemente of @.(2)
can be divided, in stany waves into fg — Vp ante Gyo By. ah
Tins = Me), © JP Egh, Per ene parse Te kx ually” enough %0

Torte prsjetve moriy, e Csoniocl Ch 2 ar Gali os
PBL waa avsniac atest JOURNAL, JANUSEY 19H

examine only oa of thee sets fr example the frst ine i the table

‘A subspace of Gy(g) ic elled am atplatet oer ig) and 8 mambers
exe callers, Tre lng of Teter ism nd the sume of roazero
oondinae ins letter i ie weight. Kory alae contain the ltrer
LUM) The ings weiter es i ented by wed
is alo wollte ph of thea. Th diego he nie
tea veotar space seer Pig} isk By @yln) we mean an alphaber @
‘ich dimension B, weight an length (ofeach lectern. For exemple,
fina) ll

"An alphabet (isin) contains ¢ lotr, Srom whieh wo pik any &
indepeniont veewrs as generators. We waite thes as the rows of a
fx sn mattis ACG), the geuraor mab of &. For example,

(1)

Fe eyo mato au (223). Wo ny ame a et
fa gear mate ctl etry of sonore ha Tp
Steph fo& bp of a
Sarde el tT lets eg), wot ll ae or enol. ¢
colt s gone mrs) be eae be ene
Poi ofa pect sue Te), uf pete neo (k — 1)
{ror F(g: Weal sayeth oresson th ne we sere
pont Tq) tole tien cum vector eg
few
a.
1 sg) oni — 1469 1 pinta PH, Qa AQ ae
hate Th tn of) ate era epoence
sit tei pcre oh cin
Tt wt the goers ae a

WB @
Rie te oe
MEO) = Re Joe ge te

Ralt e o e
rurreisunvub wine coRRFETMa chore 285

tad esi te rows Fy, os > Re and sho columns Qs, Qyy ==> 5 Qe
“Rngacd the column re of pointe 2 7}. ‘Thre are xa" &
jnceperden eolumrs. so Thi 9208 pats spar the xpuce Tafa). et
fsbo tho nurshor ol irs which some maltiple of she eolaran Q, [tbe
"ripe being cement 07 F¥9)| appear in Si. Tho comespond-
ing point in T'tg}abal the hare muplty ,. We enn now Sto.
‘ave Rose's thecrem,

oe

om fs Lat

hou zeta pot a Ph stg) Let S denote the at of points Qy Qe o>
eh conttod with proper mulls. Then the weight of the Vetter

HO) = Al bey = oo Maley CPG
oF @ equal tothe number of ints of she et 8 which dont Teo the
ayperplane
HO) Sy Phat tte = 0
oP. eh
Prat It for example,

aga ag E00 Dg =
sho point Q, esos: H(A), ‘he soroa im the Inter RA) avi rom the
points 0° 8 vhih Tie on 220), td he mter of sores wil be the
tnber af seh pants ernnted with proper lip. The weight of
Fah) se numberof fo nonaten neat, whieh tho numlr of
pints of (aga eountel with prper multiply) not Wing on HC},
‘This proves the theorem,

Tn Fig 1, the projective plane 1,(2) ix over the Geld (0,1). Note that
2.040, a sao eines

Teen poof th oes the
28 ame ms, store mneesT IT, MANCHA GL

olde rin Hig, 1s the vet wo obtain w gnorator

Mond

fa alae E254). Hise Fn egw hae sw a6 a
Teast tv point 02 not on ay in af 223

Tr thi eon ele up the ution of equivalent apabels, ana
show how Sepian’s definition of equivalence may be extended to the
ore general ase. ist we discs what properties one would inttively
Tope for fon wah @ dein

"We may onsideran alphabet ae an aray of letter armanged ene under
nother in such a way that we ean speak o te columns. We ksow that
the opetations of permting the eakimns,malipng any eon hy an
tlernent of 8"(q) sa tern Use ent te en ale
‘ello change He erie corr presto of the slpbabel. The del
nition of equlvalenee between alphabets ahnuld allow u2 to do as maeny
of these things as posse

it Bove orem we ral hat the waight of every Inter of a
alphabot i determine! by che properties of ae: of pints in 7-9)
‘Post we wish that all alphabets desved fre the aura ae: of poitss
sion be euvalen; secondly, ito arte a pote SS" hove, i some
‘mie, the sat incidence elstione with the byperpes of Te9(g) they
should give Hise to equisulene alphabets
runreaauvbt eeuor-connsrnnc copes 27

tiven a st af points 8 in Ty ag), e dorve an alpha from ha
by means ofa generator matrix. We obeain the gonenr mite by the
calls sep

1 Fis a enna system in Tht)

2. Write Une coon!vmtes of the poizce of Se eulumnue of a manic
repeating enc rien {ime nereetsly myncr atively + with the proper
pity

"The ont i wish se write te wala ie iar alo if is
socha column ye have testing of ng MK, A 6 F(t Tess
iis appaceat That a great may ere renator mateoee may rie
fro the se se of pies

“We sal presealiy give tpasute ilsincie delntions of equialenr
between no sets of pints, two trons and tee alphabets, ad shin
fw they ane ineerzeate, Trt we give a brief desription of she rn
Tineason arp of Pt -4

‘lnetion snag of the set of points of TC) ont ise
swish preserves al nese propersin: a ite five ita es
loves inta ples, ines sunnah agent Sat Tee through point and
fons The eolineations o7s-429) ema n grap, devoted by C2.
1 eusingalor lies” roirsive.ranscormtot ol rutaatn ie a
Celine iene the {aor transition of eoodineteendaced
tor anrmatimnngtists ofthe grows teal Fa) Let Pe rhe gro
cf Hnour projective tramsormatinns, aed aCe: Bye group of tars
Klein ions inca ystems of Ue gece eld. Then ane
rolneaion of Cog! ean Iw expeesed tthe xl of member of
Petyp aid a cies of tf [Alton a elemens of Peg docs
train senoral somite sith at element), the evo roups
Frommite as sump 1g] We real tha an aitomserphisn of
Finite eld of yy lenente ie vave of the form 8 > 8, whore @
ia primitive clonht; and fo woutevial nulomorphism, 0. >< m,
‘Phe gers othe fal (the laments he pene subield ace nod
hengal by sic mapper pez ed haw rvs al
orgies and rin awe CGp) — PEE)

‘Wenn she the flue ern of equivalence

Dosactine 1 The "vwendesee so a pots 8.8" ane equiva i
ther exists wlinnsion af The) whic sens 8 ato SY. We mnie
sees

1) oo contin tea

Welsh thom encoing tho beep
288 nuh uu sven moeuteAt soUKAL, saneaat ORL

Definition 2: "Two (8 Xn) generator matsons HEMT over PCG) ave
cuvalent i

AP = pb, ath = A.
Here is an anton 0° ie grea! field ppd toh eis in

FF gat inverle % wostix over qr an Xn) pela

sats, Ao (uonsingulee} dingane: (x X 1} mattis ov
Sine has only we onsen eniry ix euch row and column we ot

ass chavee A Un

Dentin 8: Two slphabrs @ und @ are equivalent if thor wits
ote then a weight pacrsing sen homnphim,

‘Asomisbonorphinn # betwen tons voor apes, ie aay
pried by deseibing shot Iappens ta the ae vesters By ++ He
Sf, tal easing an antomwnptns of the groin Del The mapping

FOR My bd
H (San) =Zecont

inva sembisemorphism provided that RC, ++ Rave Lien indeple
‘oly any seatomogphizn em be decribed in this way

‘We note ako thot & weighl-preverving rapping ofan alphabet & once
au alphabet is necessur ano to ue for only letters of 200 weight

in ean map onto the oro (0 --- 0) wf @
nal of thew dinitnny xuivalener a ix usta properties ey

is symmetrin, flexion nd Imei
We now show thet the shes destin:
use te be made preese,

compote; than

Definition 1- > Deion 2,

Definition 2 Defines 8,

fn — Definition 1

‘Tiewron 2 A 8,8 ae aguvalont in the aense of Definition I, th Ve
sacrigs JY, to whieh thy givers n a fed coordinate eaten, are
cquivalent in the sense of Debut

‘Povo Lct § bo an ondesing of the st S, and S' the ordering of S*
sl tna
Tet 34(8), M(S%} deme she mateoes with cola
ee

“Then thore exist ponanation matries eh tht
My! = MiSs = goTi83, MES} = ae

AP = ght, AP = ate = 3,

olla rE 6 4 perraion mats
"Pheoroos tthe generator nears WF" ate eeuivalent in cho renee
ff Defiion 2, the tha alboers 0 dered tram them ze equivae
Tent in the eine of Deion
Prof: We have

w

eat, ata
Tbe be the alpha dove oma At, Wo st 9 welght-precerving
mnephisnt A bebare & aad Cate exh perenoving see
Tromphien f betv eo ad Wr dine Pa Folleae 1 eller
of a hen
Ma) = Hed
“This enriy» weight -persrvn mepying, snes effect i to pemaute
te entries vd wip eur em Py an element of AC). Ls
fab iar, for 8, y= He ane the sof A ane at +++ RP he
tows ob AP ee have
NUR) = Rend = Ie,

a(S) = Zastea = Eine

We deine fs flows: AEH? = rj. go yt) Hs wetter of a, ther
200 umm mea, svarme EnemTeAT oarsAn, TANCAN 19H

Hitt) = Wr) 8100,
implies r= 0
"To show that its semi-comarghiem,

lo) waht pera, sine

ea
sro ahsore that g 2M" is also» onsnator matt of. Tote
Ie the roms Mf, ade
Tote momo A, with Ro" = Gea
wwe have

aati arg) = Prat

“(Gen) -ebond Be) a

1(Zats)

= Dalene
We then hee

RRO LR, aR,

Bek tS Biel

ig Jf isa rghcpneryng, senisnmaephsnn bearaen 6 al
"Phere $ Unt 44° be egivnlnb arable he vere of Deion
S.and 32.47" be any generator matic «Ga. Tin the eoodinate ya.
teen in T(9), and let 8,8" be The seta of points whe coorinates ara
the columns off and AW" ‘Phen Sand 8" aew equiva i he ree of
Defsition |

“Eemog: Vet the xlhable eat We ceeted Uy a wl -pneer ing
APAM* are generator mates of and. sre Tak
i els of

AF aoe ME eve rise to the same (unordered) tof points