Q$.4($i$)$ In a test generation process for a combinational circuit six

tests, $t1,t2,t3,t4,t5,t6$ are generated to cover a set of given faults.

Later it is discovered that we are interested only in a subset of the

faults and the subset consists of eight faults, $f1,f2,f3,f4,f5,f6,f7,$

f$8.$Though simulating the six tests for each of the faults $($without fault

dropping$)$ we find the detection capability of each test as given below.

The test $t1$ can detect faults $f3$ and $f5.$

The test $t2$ can detect faults $f2$ and $f7.$

The test $t3$ can detect faults $f2,f3,$ and $f7.$

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The test $t4$ can detect faults $f1,f2,$ and $f7.$

The test $t5$ can detect faults $f4,$ and $f6.$

The test $t6$ can detect faults $f,f4,f6$ and $f8.$

Find a smallest set of tests that can detect all eight faults. You must

show your work to prove that the set obtained by you is the smallest

set.

$($ii$)$Consider a characteristic polynomial $x^5+x+1.$ Now answer the

following and you must show your work for full credit.

$($ii$.$a$)$Is this polynomial primitive?

$($ii$.$b$)$Give an internal exclusive$-$OR $($modular$)$ realization of this

polynomial.

$($ii$.$c$)$Design a four stage cellular automata whose cells follow Rule $90.$