Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site duke.UUCP Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!duke!aff From: aff@duke.UUCP (Amr F. Fahmy) Newsgroups: net.math Subject: A new coding problem Message-ID: <5953@duke.UUCP> Date: Wed, 26-Jun-85 10:48:18 EDT Article-I.D.: duke.5953 Posted: Wed Jun 26 10:48:18 1985 Date-Received: Thu, 27-Jun-85 07:47:19 EDT Organization: Duke University Lines: 54 Here is an interesting problem that I am sure you will enjoy working on. /*------------------------------------------------------------------------*/ The problem : Let S be the set of all binary words of length n. Let E be a set containg exactly L of the words in S. Define C(E) = sigma min diff(x,e) x in S e in E where diff(x,y) = m if x and y are two binary words that are different in exactly m bits. Of course m <= n. to be the cost of the set E. Now given the two integers n and L, the problem is to construct a set E such that C(E) is minimum. /*------------------------------------------------------------------------*/ Example : Suppose n=4 and L=4, we can construct the following two sets : E1 = { 0000, E2 = { 0000, 0011, 0001, 1100, 1110, 1111 } 1111 } C(E1) = 16 C(E2) = 12 Of course the set E2 is more desirable than E1, in fact E2 is ONE of the optimal sets I am lookoing for, you cannot find a set E3 with cost less than 12 !! /*------------------------------------------------------------------------*/ Now lets talk algorithms, the set E2 above was found using an exhaustive search. The exhaustive search is very expensive. There are some ways of decreasing the space of search but still it is expensive. What I would like is a polynomial time algorithm to solve the problem. I have worked on this problem for sometime now, I have no clues, and I do not know if the problem is NP-complete. If you are interested please send your comments, solutions, proofs... etc to aff@duke thanks a lot in advance to who ever replies back. Amr Fahmy aff@duke ..!mcnc!duke!aff