CHAPTER 3: Cyclic Codes
... We show that all cyclic codes C have the form C = f(x) for some f(x) Rn. Theorem Let C be a non-zero cyclic code in Rn. Then • there exists unique monic polynomial g(x) of the smallest degree such that • C = g(x) • g(x) is a factor of xn -1. Proof (i) Suppose g(x) and h(x) are two monic polyno ...
... We show that all cyclic codes C have the form C = f(x) for some f(x) Rn. Theorem Let C be a non-zero cyclic code in Rn. Then • there exists unique monic polynomial g(x) of the smallest degree such that • C = g(x) • g(x) is a factor of xn -1. Proof (i) Suppose g(x) and h(x) are two monic polyno ...
VEDIC MATHEMATICS : Various Numbers
... Non-constructive Proof • Show that there are two irrational numbers a and b such that ab is rational. • Proof: Take a = b = √2. • Case 1: If √2√2 is rational, then done. • Case 2: Otherwise, take a to be the irrational number √2√2 and b = √2. • Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is ration ...
... Non-constructive Proof • Show that there are two irrational numbers a and b such that ab is rational. • Proof: Take a = b = √2. • Case 1: If √2√2 is rational, then done. • Case 2: Otherwise, take a to be the irrational number √2√2 and b = √2. • Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is ration ...
