1. Lecture 15 The Minimum Distance of a Code (Section 4.4)
Theory of Information Lecture 13
Theory of Information
Lecture 15 The Minimum Distance of a Code (Section 4.4)

2. u-Error-Detecting Codes
Definition A code C is called u-error-detecting if changing u symbols in any of its
codewords does not yield another codeword in C.
A code C is called exactly u-error-detecting if u is the largest such number, i.e., if
the code is u-error detecting but not (u+1)-error-detecting.
Examples:
C = {000, 111} is exactly error-detecting
C = {000000, , } is exactly error-detecting
C = {1000, 0001} is exactly error-detecting
Give examples of exactly single-error-detecting codes.

3. v-Error-Correcting Codes
Definition A code C is called v-error-correcting if nearest neighbor decoding can
correct any v or fewer errors, without encountering a tie. A code C is called
exactly v-error-correcting if v is the largest such number, i.e., if the code is v-error
correcting but not (v+1)-error-correcting.
Examples:
C = {000, 111} is exactly error-correcting
C = {000000, , } is exactly error-correcting
C = {1000, 0001} is exactly error-correcting
Give an example of an exactly 2-error-detecting code.
Binary repetition code of length n, denoted Rep2(n), contains just two codewords:
n 0’s and n 1’s. It is exactly error-detecting and exactly
-error-correcting.

4. Telephone Numbers
The set of all active US telephone numbers is a decimal (10-ary) block code of length
10. Its current size is about 115 million.
Could it be made (at least) 1-error-correcting?
I.e. is there a single-error-correcting decimal code of size 115,000,000?
At this time, the maximum size n of a decimal single-error correcting code is not
known. What is, however, known is that
82,644,620  n  100,000,000.
Conclusion: the goal cannot be achieved for US, but can be achieved for a somewhat
smaller country.

5. The Minimum Distance of a Code
Definition Let C be a code with at least two codewords. The minimum distance d(C)
of C is the smallest Hamming distance between distinct codewords of C.
Definition An (n,M,d)-code is a code of
length n (#symbols in each codeword),
size M (# codewords), and
minimum distance d (smallest distance between codewords)
Examples:
C = {000, 010, 011} d(C) = C is a code
C is exactly -error-detecting and exactly error-correcting
C = {00011, 00101, 11101, 11000} d(C) = C is a code
C={000000, , , } d(C)= C is a code
Parameters

6. Three Theorems Theorem A code C is u-error-detecting if and only if
d(C) ≥ u + 1
Theorem A code C is v-error-correcting if and only if
d(C) ≥ 2v + 1
Theorem A code C is exactly-v-error-correcting iff
d(C) = 2v + 1 or d(C) =2v +2,
i.e., v = (d(C) -1)/2.

7. Mixed Error Detection and Error Correction
Strategy (for v0): If a word x is received and the closest codeword c is at a distance v,
and there is only one such codeword, then decode x as c. Otherwise declare an error.
Definition A code C is simultaneously v-error-correcting and u-error-detecting if,
whenever v errors are made, the above strategy will correct them and if, whenever
at least v+1 but at most v+u errors are made, the above strategy simply reports an error.
Theorem A code C is simultaneously v-error-correcting and u-error-detecting if
and only if d(C)  2v+u+1.

8. Maximal Codes
Definition An (n,M,d)-code is said to be maximal if it is not contained in any larger
code with the same length and minimum distance, i.e. it is not a subset of any
(n,M+1,d)-code.
Are the following codes maximal?
{0,1}
{00,11}
{001,111}
{0000, 0011, 1111}
{0000,0011,1100,1111}
Theorem An (n,M,d)-code C is maximal iff, for all words x, there is a codeword
cC such that d(x,c) < d.
Is {001,111,100,010} maximal?

9. Pros and Cons for Maximizing Codes
Advantage: The code, after enlarging to a maximal one, will still remain (d-1)/2-
error-correcting.
Disadvantage: While the old code may be able to correct more than (d-1)/2 errors
0ccasionally, the new code will not be able to do so.
Is C={00000,11100} maximal?
Let C’ be any maximal code containing C
If is sent but is received, can C correct it?
can C’ correct it?
why?

10. Theory of Information Lecture 13
Exercise 1
Consider the binary code C={11100,01001,10010,00111}
Compute the minimum distance of C
Using minimum distance decoding, decode the words
10000
01100
00100

11. Theory of Information Lecture 13
Exercise 3
Consider the code C consisting of all words in Z2n that have an even number of 1s.
What is its
length?
size?
minimum distance?

12. Theory of Information Lecture 13
Exercise 9
Does a binary (7,3,5)-code exist?

13. Theory of Information Lecture 13
Homework
Exercises 2 and 8 of Section 4.4.