# Homework 4

Due: Monday, October 18, 1999

• Reading Assignment
• MacWilliams & Sloane, Chap. 7
• Peterson & Weldon, Chap. 8
• Berlekamp, Chap. 5

• Hints for Homework 4

• Problem 1.
Compute the addition and multiplication tables for the ring

R3 = GF(2)[x]/(x3+1)

Also express each of the following ideals in the ring  R3 as a set of elements of R3 .

(0),  (1+x), (x2+x+1), (1), (x2+1), (x3+1), (x5+x+1).

For example,

(0) = { 0 }   and   ( x4 + x2 +1 ) = { 0, x2 + x+1 }

• Problem 2.
Let V be a binary linear code given by the generator matrix
```			[  101011  ]
G =  	[  011110  ]
[  000111  ]
```
• a) Find a parity check matrix H of V .

• b) Construct a maximum likelihood decoding table for V .

• c) Use H to reduce the maximum likelihood decoding table of b) to an error/syndrome table.

• d) Demonstrate how your error/syndrome table can be used to decode the received vector r = 111101.

• e) Use the generator matrix to create a list of all code vectors of V. Then use this list to determine the minimum distance of V.

• Problem 3.  Let  ksi  be the primitive element of  GF(26)  defined by  ksi = x mod x6 + x + 1.  Compute the orders of the elements of  ksii  for  i=0,1, ... , 62.  Summarize your results in a log/order table.  For which i's are the ksii's primitive?  Do you see a pattern?  Make a conjecture about this pattern.

• Challenge Problem 4.
Devise a procedure for correcting double erasures for the Hamming [2m-1, 2m-1-m, 3] code.

Last Modified: October 11, 1999