1. Consider the following system of equations with coefficients in the field Z7:

Question

1. Consider the following system of equations with coefficients in the field Z7: x1 + x2-2x3 + 4x4-2x5 = 0 650 Let C be the set of all solutions of this system in Z (Note: a vector (81, s2, 83, s4, s5) is a solution of the system if the substitutions xi = Si (1 equations in the system being satisfied). i 5) result in all (a) We can view C as a code of length 5 over Z7. In at most two sentences, explain why C is linear (Note: Repeating the definition of linearity is not a valid answer) (b) Find (with justification) a generator matrix of C. (c) How many codewords are there in C?

Explanation / Answer

there are 5 variables and 3 equations so we will take 5-3 arbitary variables let x2=k ,x5=l then

from eq 3 we have x1=3k+6l

on substituting this values in eq 2 and eq 1 we will get the vaector

C is linear all the variables are with only power 1

here cofficients are -2,4,6,3,5,1 they will form cofficient matrix under modula 7

-1 is equal to 6

-2 is equal to 4

so we have only 6,4,3,5,1

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