# Coding Theory : Linear Codes

Please see the attached file for the fully formatted problems.

(a) Explain what is meant by

(i) a linear code over Fq

(ii) the weight w(x) of a vector x

(iii) the weight w(C) of a code. Prove that,...

(b) Prove that w(C) = d(C) if C is a linear code.

(c) Define F-linear equivalence of codes. State the three row and two column operations which may be performed on generator matrices to obtain F-linearly equivalent codes, and prove that these operations do preserve F-linear equivalence.

(d) Let C be the linear ]F3-code with generator matrix

....

(1) List the codewords of C.

(ii) Find d(C). How many errors can it detect? How many errors can it correct?

(iii) Prove that C is a perfect code.