Coding Theory : Sphere Packing and Coset Leaders

Explain what is meant by the sphere St () with centre i and radius / t in the vector space F. Show that...
Let C be a linear [ri, k, dj-code over Fq and set t [i]. Show that...
for all distinct elements 7 and of C. Hence show that...
Give the definition of a perfect code.
Give the definition of a coset leader/'Let C and t be as in (ii), and let L denote the set of all coset leaders for C. Prove that S...
Prove that if C is perfect then ....
Using the fact that the binary code C with a generator matrix
1000111
G 0 1 0 0 1 1 0
0010101
0001011
is perfect, find the coset leader with syndrome (111).

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