Question

Problem 4. Is there a perfect q-ary (n, M, d)-code with the following parameters? Explain your answer.

(a) q = 2, n = 7 and d = 7.

(b) q = 3, n = 10 and d = 4,

(c) q = 3, n = 10, and d = 3.

Answer 1

Answer:

Given that:

Is there a perfect q- ary (n,M,d) - code with the following parameters.

Condition for code to be perfect is:

its n,m and d values satisfy the sphere packing bound.

i e.

With d=2t+1

(a)   

q =2 , n=7, d=7

d=7

  

  

= 64

   (for perfect code )

M(64) = 2⁷

^{M =128 /64}

^{M = 2}

^{(b) q=3 , n=10 , d=4}

^{2t+1 = 4 (i, e}

2t =3

-t =3/2

t=1

= 1+20

=21

no, perfect code cannot be derive from

(c) q=3, n=10, d=3

2t+ =3

2t =1

t= 1

= 1+20

= 21

  

Not give a perfect code

(because , we don t have integer value of M that satisfy above equation)