1 Problem sheet 10 2006

MT3610/5461 Error correcting codes

Ex. 1 Show that the Latin squares L_{1} and L_{3} of order 4 are not orthogonal. Construct two orthogonal Latin Squares of order 4 as follows: Construct the field of 4 elements by taking all polynomials with coefficients 0 and 1, and reduce modulo x^{2} + x + 1. Show that the 4 elements are: 0, 1, x, x + 1. Write down the addition and multiplication table of this field. From this find two MOLS of order 4. Bring them into standard form.

\begin{matrix}Ex. 2 \\ • \end{matrix}Show that f(x) = x^{3} + x + 1 is irreducible over the field with two elements.

• Write down all 8 elements of the field 𝔽_{2}[x]/(x^{3} + x + 1).

• Show that x is a primitive element, i.e. that the x^{i} generate all nonzero field elements.

• From this describe the Hamming code Ham(3, 2).