**Coding Theory**

** Third Assignment: due Fri. Mar 9**

- CT 3.1.11 codewords with k zeros (over an arbitrary field).

- a. Find a code with minimum distance
* d = 5 *, redundancy
* r = n - k = 8* and length *n * as large as
you can.

b. Compare your result with the Gilbert-Varshimov bound.

c. Do the same for *r = 9 * if you want to impress me.

- CT 3.3.10, decoding for a Hamming code.

- CT 3.4.4, SDA for decoding an extended Hamming code.

- CT 3.4.5, extended Hamming code is self dual.

- Write down a parity check matrix for the ternary Hamming code with
redundancy
* r = 3*.

- CT 3.5.2, another generator for the Golay code.

- CT 3.5.3, Golay code is self dual.

- CT 3.6.5 f, g, h, i, decoding for the extended Golay code.

- Find the number of codewords of each possible weight, 0,1,..,24
for the binary extended Golay code (CT 3.7.7-9).

- a. What is the difference between puncturing a code and shortening
a code?

b. Use the ternary Golay code to illustrate the difference
by puncturing and shortening with respect to the last coordinate.
The ternary Golay has matrix * [I_6 A] * where
* A * is

011111

101221

110122

121012

122101

112210

c. Which of the two, the shortened Golay or the punctured Golay, is perfect?