Chapters 16 & 17
Sarah Cameron
18 March 2010
 Review
of Modular Arithmetic
 Identification Numbers
• ZIP Codes
• Bar Codes
 Binary
Codes
 Encryption
a
mod n = the remainder when a is
divided by n
 Examples:
• 3 mod 2 = 1
• 37 mod 10 = 7
 Identification
Numbers:
• Unambiguously identify the person or thing with
which it is associated
• Should be “self-checking”
 Five
Digits: ABCDE
 A: Represents one of 10 geographic areas in
the US, normally a grouping of states
 BC: These two digits in combination with the
first, identify a central mail-distribution
point known as a sectional center.
 DE: The last two digits indicate the town or
local post office, the order is often
alphabetical for towns within a delivery area
4
code: FGHJ
 FG: Represent a delivery sector.
 HJ: Narrow the area further based on the
needs of the delivery sector.
Bonus Trivia: Using the ZIP + 4 code makes
you eligible for cheaper bulk rates. 
 Binary
Code: any system for representing
data with only two symbols
 Bar Code: a series of dark and light
spaces that represent characters
 Has
been used since 1973.
 12 Digit number: KLMNOPQRSTUV
 K: This digit identifies the kind of
product.
 LMNOP: These digits identify the
manufacturer.
 QRSTU: These digits are assigned by the
manufacturer to identify the product.
 V: The final digit is the check digit.
 Check
Digits
• Used for error detection and correction
• Can be the last digit in an ID number or a binary
code.
• Often for ID numbers:
 Sum the individuals digits and mod by the check digit
Take binary string a1a2a3a4 and append three
check digits c1c2c3 so that any single error in any
of the seven positions can be corrected
 Choose 3 different sums:

• c1 :a1 + a2 + a3
• c2 :a1 + a3 + a4
• c3 :a2 + a3 + a4
If the value of the check is even, the check digit
should be zero.
 If the value of the check sum is odd, the check
digit should be one.
 Using this method you are able to produce a list
of all possible a1a2a3a4c1c2c3 combinations.


Nearest-Neighbor Decoding
• This method decodes a received message as the code
word that agrees with the message in the most positions

Distance Between Two Strings (of equal length)
• The number of positions in which the strings differ.

Example:
• Real Message: 1000110
• Received as: 1010110
• The distance between the real message and the received
message is one.

If there are two possible translations for an
erroneous message, it is not decoded.

Binary Linear Code
• Consists of words composed of 0’s and 1’s obtained from all
possible messages of a given length by using parity-check
sums to append check digits to the messages. These words are
called code words.

Weight of a Binary Code
• Is the minimum number of 1’s that occur among all nonzero
words of that code.

Accuracy:
• If the weight of a binary code is odd, the code will correct any
(t-1)/2 or fewer errors.
• If the weight of a binary code is even, the code will correct any
(t-2)/2 or fewer errors.
• If you simply want to detect errors, the code will detect t-1
errors
 Encryption
• The process of disguising data
 Cryptology
• The study of methods to make and break secret
codes
 Data
Compression
• The process of encoding data so that the most
frequently occurring data are represented by the
fewest symbols
VW
U
XZ
Y
B
A
C
T
D
S
E
R
P
Q
F
O
H
N
M L
K J
I
G
1.
Delete all occurrences of
h and w.
2.
Assign numbers to the
remaining letters as
follows:
•
•
•
•
•
•
•
3.
0: A, E, I, O, U
1: B, F, P, V
2: C, G, J, K, Q, S, X, Z
3: D, T
4: L
5: M, N
6: R
If two or more letters with
the same numeric value
are adjacent, omit all but
the first.
4.
Delete the first character
of the original name, if still
present.
5.
Delete all occurrences of
A, E, I, O, U, and Y.
6.
Retain only the first three
digits corresponding to
the remaining letters.
7.
Append trailing 0’s if fewer
than three letters remain.
8.
Precede the digits with the
first letter of the name.
A B C D E F G H I J K L MN O P Q R S T U V WX Y Z
D E F G H I
J
K L M N O P Q R S T U V WX Y Z A B C
• Used by Julius Caesar to send messages to his
troops.
• Decode “DWWDFN DW GDZQ”
• The encoded message: “ATTACK AT DAWN”
ATTACK AT DAWN
0
19
19
0
2
10
0
19
3
0
22
13
MATHMA TH MATH
12
0
19
7
12
0
19
7
12
0
19
7
MTMHOK ?? ????
12
19
12
7
14
10
• Choose a key word, which can be anything.
• Add the digit position of the original letter to the
digit position of the key word letter.
• Mod that number by 26 = digit position of the
coded letter.
• Repeat the key word for as many characters as you
need.
• Answer: TA PAPU;19, 0, 15, 0, 15, 20
 Questions?
 Applications?
 Homework: (7th
• Chapter 16: #74
• Chapter 17: #11
Edition)