q-ary code proof using spheres
Show that there is no q-ary code of length 5 and distance 3 with more than codewords. You should prove this from first principles using spheres and without using known bounds on the number of codewords.
This is my working so far:
First prove that there is a q-ary code of length 5 and distance 3 with exactly codewords. consider the spheres of radius 1 centred on codewords. Each such sphere contains 1 + 5 = 6 words. Since the code has distance 3, no two such spheres intersect. Otehrwise we would have a pair of codewords at distance 2 or less. Since there are codewords, there are distinct words in these spheres. However there are only distince q-ary words of length 5. therefore we have to show that
= for some q that is a natural non-zero number.
i worked out q = 2 therefore such a code exists. but now im having problems proving that no q-ary code with MORE THAN codewords exists. also, with q = 2 the no of codewords = can anyone tell me why this isnt a whole number?
PLEASE HELP!!! any help or advice would be GREATLY appreciated!!!