Question

encoded À (1.4) code. [m ma m3 ma] [m, ma M3 M4 mi ma m] (1) what is & matrix ? (t) What is it matrix? is what is dmin!

this question is about the coding theory. Please help me for this problem with explainations, thanks for your help!

Answer 1

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tas URE ERROR, CONTROL CODINGO (LRC) LINEAR BLOCK CODES ( Hamming codes) combined objective of channel encoder and decoder is to detect and coerentes & Channel encoder divides binary sequence isto sequential meuage bloches each having k bits in it Each of gk distinct message blocks will be transformed to a renique colle word each having 'n bits is it. a Error fie Channel 10100. k/k к] channel decoder Encoder binary Segi Inbits kink) 2020-3-14 11:06

petations 0 Addition (modulo-2) orlx-OR 000 = dition (modulo-2 . multiplication 0.0= 0 0 1 - 0 1.0 = 0 1.1 - 1 100 = 1 1 - O Hamming weight and distance: Hamming weight ga member of i's in it. - Hamming det imming weight of a code word specifies member of positions in which a code words differs. soring distance specifics eg:- C = (1010) 2 wasta d (c,,.2) = 3. 2 = (0100) I w(Ca)=1 Minimum distance (dmin): - A specifies Minimum It specifics Minimum possible bamming distance between any pair of code words is called as minimum distance (dmin). Smallust possible bamming weight of nonzero code word. will be equals to domini 2020-3-14 11:07

V kumen JENERATOR MATRIX (GUS- Ket message block d .= (doda.....du) ik bite) Code word c = (ciola...ck . Chtm =d, 1 message bits man-k pwity bets. C = de nk) parity bits. ch- du Cl+1 = Pid, Riada o pikor I 'P'~ parity matrix ( my A Ck42 = Pood, Paad. Pa 3 do 6 ... dakde kim - Prosd, Pmada ... Bankar. GedGro=dG 10 nxk kan 7110 [6., C2, Cz..... Ck Cheti ... Cum] = [d, da .... del [a do... dal lol -- 0 100 - 01-

a C2, C3 .... a ..... Chtml= ... Chtm]= Lara [d, da ... du] OD- P, B ... Pri B, 2 Poo ... PM2 O-OO... Pik Pak Pook PT So, kxn. Pkxm kx Toor kxth-k) گیره؟ Ikxk 1 G = [I: PT] P [ Pn Pia ... Pik ] P21 P22 ..... Pak LPmi Pm2 - ... Pak mxk or 7 (ook) xk C = [K; N-k] = G = [ Ik :p/kxN. c= [N-K; k] = G = [PT : Ik] kxN. bobe Cootenay abera Note : - above LBC is called as (oik) LBL 2020-3-14 11:08

teo abou Corect up to (6,3) LBC hard, each xe PARITY CHECK MATRIX :-/(H) po 2 exerat. 3 dmin3, can detect Lupilo 2 bxdande each received de coord. -Too-Obe c Cute =[ Ik : pl]xxn H= [P: Im] (onkoxk (5k) xo : KOLOR LE Order : H (-K) x Toonnaureus are suite om ong to rev: e generally taon Berylium, coppe Phospher-Bror I stainless-ster HT= [27] G.HT= [IK: PT] Im Inn = POP G.HT = 10l kx10-4) (C.HT = dGHT ICHT = 101x04) PSULE su REs where ATIC PRESS ACTIVE T ZOELEC X DX (1-1) 4 Summary f whi 0 C = (K, O-k) — 0 c = (n-kok) G = [Ik: P etl ovom G=[ pT: Ik] 8 H = [Im :P] Ln-k) xn KX) 2020-3-14 11:10

it (7,4) = (0,k) form, no tength of code 7 h k = = ko length of messige bit manok number of parity bite wed. D 4 We know, code (C = d.G also, G={Ix P xxn. & Six Identining madalin) Z p- paxity chook Here, c=1c, C, Cg Cy es Caly] message bids Let Logic fox parity bits be, cizdi Css d, da cas de C6 = d, dede cs = da eta dada : ce dy just writing the coefficients. P = T 0 0 1 1 0 0 1oool 0100011 ooloilo oooooo 1 XL рт D H2 Partly check matrix Mathematically, G-T IK : PT xxn H=[P: IN (MK) Yk (m_kixn H=11010 Too 1110010 100.00 From

Come titeket xesalta c lc, ca ez les es ce 4] rome or Imi mamy my hed, distance donin is the minimum possa between any pair of code ring Smallest possible hamming weight of no will be equal to domin. & pair of code worde. f weight of non-reco code word equal to a let us take (6,9 code. n=6, k 3 , m - 3 aet d = (110) u p-lelling 2 above G logic i = ans Conde comidas c= dG = [io] [G ] - T1 10] 1o ololoi oollllo | Weight (WCC) a noi of is in code 0 ю So, from above defination т т c= d. G OOO OOO oo1 110 olololl 1011 1101 lloo 11 1 1 001 10 100 Il 11 l olo 011 TOO ) dmin smallest (wa) и

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