Arithmetic modulo 2 Arithmetic modulo 2 mod 2 is like ordina

Arithmetic modulo 2 Arithmetic modulo 2 (mod 2) is like ordinary integer arithmetic, except that any answer is divided by 2 and only the remainder is kept. The rules are: 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0, 0-0 = 0, 01 = 10 = 0, 1-1 = 1 The only \"unusual\" rule here is 1 + 1 = 0. Integer arithmetic demands that 1 + 1 = 2. In mod 2 arithmetic we divide the answer by 2 and keep only the remainder: 1 + 1=2 divided by 2 is 1 with a remainder of 0, so in mod 2 arithmetic, 1 + 1 = 0. The same commutative, associative and distributive laws as holds for integer arithmetic also holds for mod 2 arithmetic. If you\'re interested, you can read more about modular arithmetic here. https://WWW.khanacadfimy.0r6\'C0mpUtin6/C0mputCT-scicncccryptography\'\'modarithmctic/a what-is-modular-arithmetic Problem: Work out: (a) 1 + 1 + 1 + 1, (b) 1 + 1 + 1 + 1 + 1; (c) 1(1 + 1); (d) 0 - 1 + 1-1-1; Error Control Coding Digital cell phones communicate using the language of computers, with voice and data sent as groups of information bits (zeros and ones) over the air Wireless communication is very difficult, the signals are reflected off buildings, absorbed by trees and rain, and scattered by street signs before being received by the antenna on the phone. Because of these impairments, it can be difficult to receive all the information bits correctly; some will have invariably flipped from 0 to 1, or vice versa. On a voice call, the effect is digital distortion. On a data call, the effect changes the data present in the transferred files. One primary reason that digital phones are so powerful is that phone designers can use error control coding to ensure that your phone call is clear and error free. In practice, error control coding adds extra bits (called parity bits) to the information bits so that errors can be corrected. The process of encoding the information by adding these extra bits, and decoding to detect and correct any errors, is called error control coding. Let\'s see how using matrix algebra to perform encoding and decoding makes your phones calls clear and error free. A set of information bits plus parity bits forms a cock word, and the set of all possible codewords forms a cock Below, we show how to encode 4 information bits with 3 parity bits to form 7-bit codewords. This is known as a (7, 4) code 4 information bits can be thought of as a matrix row of length 4 containing zeros or ones. Examples are [0 1 0 0J or [1001]. etc We shall frequently abbreviate this as 0100 and 1001, etc Problem 1: Using the abbreviated notation, list all possible 4 information bits (7, 4) code A pair of matrices are used to encode and decode the (7, 4) code Here we use

Solution

PRE-REQUISITE-ARITHEMATIC MODULO 2 :

(a) Based on given rules, we have

1+1+1+1 = (1+1)+(1+1)=0+0=0

(b)1+1+1+1 +1= (1+1)+(1+1)+1=0+0+1=0+1=1

(c)1.(1+1)= 1.0=0

(d) 0.1+1.1.1=0+1=1

Answer