Homework 1

Due: Monday, February 9, 1998
Peterson, Wesley, and E.J. Weldon, Jr., "Error-Correcting Codes," MIT Press (1981), pages 1-10 & 32-35.  This book is on 3 hour reserve in Kuhn Library.
• Problem 1.

• Form a maximum-likelihood decoding table fot the binary code consisting of the four code words 0000, 0011, 1100, and 1111, assuming the binary symmetric channel.

• Problem 2.

• Over GF(3), put the following matrix into echelon canonical form
```                        [  0 0 2 2 0 2  ]
[  2 2 0 2 1 2  ]
[  1 1 2 0 2 2  ]
[  1 1 0 1 2 1  ]```
• Problem 3.

• Suppose that the set if all received messages  x, y, ...  is  S, and that a metric function  d(x,y)  is defined on the set  S.  Suppose that transmitted messages are in the same set.  If  x  is transmitted and  y  is received, an error of magnitude  d(x,y)  is said to have ocurred.  A code  C  is a subset of  S, the idea being that in using a code  C, only messages  x1, x2, ...  in  C  are transmitted.

(a) Show that a code  C is capable of detecting any error of magnitude  d  or less if and only if the distance between messages in the code  C  is greater than  d.

(b) Show that a code is capable of correcting any error of magnitude  t  if the minimum distance between code messages is greater than  2t.