Guaealized Hypercube on a 2K vertices of degree d K

Definedby gc ga E Oilk

mod 2

Has adj equals µ EffySits

Cif l 2x e c Eod

KS entries in It a Binod 2construct GI beef

Mis d 21EH whee H Exit to

Hammyweight

µo d it IGN Head freak Is totheir graph is an 2E expander

will see AE O is a cat this Golds with deck

Coding to send message in QR transmit 0,13 nom

Darity bit be bin C Eod appeal ban Effi nod 2

Can now defect oneemer

Bat can't figure out where it is

1 error correction Hanning Code

will trasurit bs bs.bg by e s l 0 to

parity kits b D tbs 157 l

b 133 tbf tbf 0

by bestbothy 1

These satisfy

enU

If cae bit flipped to create C say Geo

then Mc the column f fit 6

and is 6 in binary So find location ofone error

Code is a uegpiyc.fi FI n m

Cbl is a codeword

tale c myHammy dat laced la Cal

minimum distance is min distance DCota

can correct up to dk errors

If dat Cc et E then fall C

dat e E e data t t dist ft E c Et distCeEldata E dz

minimum relativedistance F Insequenceof codes GG is asymptotically good if

1 Cil t O failed o e i

nCCil should grow

New Code Cts Gb for some C c HIM

As GCb tbd G bit Cbe fly Gtx mad 2

C a codewords S Cc Ca a codeword

ouch dat ffin GH

Ce Et cast beat

will see a radon G is good

R CE has mandated z I Izproof For C random Gb is Reedon A eoeryb

toprIEH i3 C.IREIEHedff.dz

prfzb tos.fr fisted E2m fC a

Analysis Let HER plgap CtDlosIBH

f znHCP

And for B HAI 2 7 O

HOTso 2 II Fl o if C t Its

That is if choose C E FI at radon

expect min rel dat f akee Hoy n l t

F 8 4

For d E T is small but 70

For generalized hypercubealso need wat oddest

orlyafadrzonpobability nesk.si

Constructions

Reed Solomon is over Fp numbers modaco a

primemessage is f fu encode by

setting CH fifi for teffp

transmit Col Cd QCp c so a P

ten If 0 has degree em ard is 0 at

our field element then It a O

The Min dot of Rscode is p m

proof let Q and Q2 be two poles of degree em

So is Q E Q Q2If I Q Reen Q is non zero

So Q is zero at most m times

and I Q 04 datCQ a p m

But Then is not over zThey l

Forney Goose Cima Fp Fz IF2

set 2lzp Acode

If men dist of Cine d

Miu dat of whole dlp m

How get CinnaAt random

By brute Searl

Justesen use all