# Coding Theory : Vectors and Generator Matrices

Please see the attached file for the fully formatted problems.

1(i) Explain what is meant by

(a) a linear code over Fq,

(b) the weight w(u) of a vector u and the distance d(u, v) between vectors u and v.

(c) Define the weight tu(C) and the minimal distance d(C) of a code C. Prove that w(C) = d(C) if C is linear.

(ii) Give the definition of the dual code C-'- of a linear code C. Prove that C-'- is

a linear code.

(iii) Given a generator matrix C in standard form for a linear code C, describe a generator matrix H in standard form for the dual code C-'-.

Let C be a linear code over F and let H be a generator matrix for C-'-. Prove ...

(v) Let C be a 3-ary code with generator matrix

...

(a) Find a generator matrix for C in standard form.

(b) Find a parity-check matrix for C in standard form.

(c) Determine d(C). What is your conclusion about the error-correcting and error-detecting powers of the code?

....

1 (i)Exp1ain what is meant by a linear code over Fq, the weight of a vector and the weight of a linear code.

(b) Let e F. Prove that

d() =w(?j).

(ii) Prove that w(C) = d(C) if C is linear.

(iii) Let D be the linear 3-ary code generated by the matrix

102

011

(a) List the codewords of D.

(b) Find d(D).

(c) Write down a standard array for D. Is it possible to give an example of a received vector with one error being correctly decoded?

(iv) Let C be the code obtained from the code D of part (iii) by adding an extra parity check: be the linear 3-ary code generated by the matrix

1 0 2 1

0111?

Show that C corrects single errors.

...