University of California San Diego
ECE 259A: Solutions to Problem Set #2
n 1. The set of all the syndromes s is equal to 2 k which is a vector space of dimension n k. Thus nk the image of 2 under any linear mapping L s cannot have dimension larger than n
University of California San Diego
ECE 259A: Problem Set #2
1. Recall that the syndrome s is a linear function of the error-pattern e. A linear decoder finds its L s , where L satisfies estimate e of the error-pattern as a linear function of s. That is e
University of California San Diego
ECE 259A: Problem Set #4
1. How many binary linear cyclic codes of length 15 are there?
(a) Prove that all the codewords c 0 , c1 , . . . cn 1 satisfy c0 c1 cn 1 0 if and only if x 1 is a factor of g x . In particular, f
University of California San Diego
ECE 259A: Solutions to Problem Set #1
1. Suppose that a vector y corrupted by at most errors and erasures was observed at the channel output. Delete (puncture) from both y and the positions where erasures have occurred,
University of California San Diego
ECE 259A: Solutions to Problem Set #4
1. A binary polynomial g x generates a linear cyclic code of length 15 if and only if it is a divisor of x15 1 over GF 2 . Using the factorization
1
x
x
x
we conclude that there
University of California San Diego
ECE 259A: Problem Set #3
2. (a) Prove that f x x3 x2 2 is irreducible over GF 3 .
(b) Let denote the root of f x , and assume that f x is used to construct GF 27 . Compute 2 1 2 2 in GF 27 . (c) What are the possible m
University of California San Diego
ECE 259A: Solutions to Problem Set #3
1. The order of must divide 2 9 1 511. Hence it is either 7, 73, or 511. Clearly o 7, 7 since otherwise is a root of x 1 which contradicts the fact that its minimal polynomial has de