**Error Correction**

(Geometric structures)

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

1. Compute the Hamming distance between the following strings of the same length:

a. 11110011 and 10101100

b. 11001111 and 11001100

c. ACGTTAG ATGTATG

d. TAGTAG AATTTA

e. 112233664455 212112233455

f. ++++***++ ++****++

(What is the "natural" alphabet for each of these problems above?)

2. An 4x4 image is coded with the error-correction code:

A = 111111 B=111000 C = 000111 D = 000000

If the image is encoded from top to bottom and left to right what does the image look like if no more than one error per code word is made (the bars help separate the code words but need not be present):

1111111\111111\111110\000000\111000\011000\000100\000010\

0000111\010111\111000\000000\111011\0111111\010111\000100\

How many binary strings are there of length 6? How many of these are within one unit of hamming distance from a code word? What can you say about other error detection or correction aspects of this code?

3. Suppose one draws the 4 point affine plane with 6 lines.

a. Write down binary sequences of length 4 that arise from indicating which points are on a given line (i.e. form a sequence where the ith bit is 0 when the ith point does not lie on the line and 1 when it does).

b. Compute the Hamming distance between each pair of the 6 binary sequences you get.

c. Can these six sequences be used to form an error correction code?

d. Construct the 6 binary sequences which one gets by interchanging the roles of 0 and 1 in the 6 sequences above. Compute the Hamming distance between these binary sequences. Can these sequences be used to form an error correction code?

e. Can one add the sequences 0000 or 1111 to the above sequences to any useful effect?