Paper 3, Section I, G

Coding and Cryptography
Part II, 2011

What is the rank of a binary linear code C?C ? What is the weight enumeration polynomial WCW_{C} of C?C ?

Show that WC(1,1)=2rW_{C}(1,1)=2^{r} where rr is the rank of CC. Show that WC(s,t)=WC(t,s)W_{C}(s, t)=W_{C}(t, s) for all ss and tt if and only if WC(1,0)=1W_{C}(1,0)=1.

Find, with reasons, the weight enumeration polynomial of the repetition code of length nn, and of the simple parity check code of length nn.