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