Digital Logic Lecture 4

Binary Codes

By

Ghada Al-Mashaqbeh

The Hashemite University

Computer Engineering Department

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Outline

Introduction.

Character coding.

Error detection codes.

Gray code.

Decimal coding.

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Introduction

Binary code is a sequence of 1 and 0 to represent specific values or quantities.

Simply it is a substitution or a simplification technique.

Binary code objectives: Security assurance: encryption and decryption

techniques used to secure communication and data storage.

Civil applications: such as text coding in computers, and codes used to reduce digital communication errors opportunities.

Mainly we are interested in civil applications.

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General Rules of Binary Codes

For n-bit binary code, the number of values that can be codes (or simply the max number of codes) = 2n.

The min number of bits required to code m values = ciel[log10m/log102].

The max number of bit used to code m values can be >= min limit computed using the above equation (i.e. there is no restrictions on the maximum limit of the used bits).

Each value must be assigned a unique code and vice versa to enable correct encoding and decoding.

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Binary Codes Types

Character coding.

Error detection codes.

Gray code.

Decimal coding.

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Character Coding -- ASCII

ASCII stands for “American Standard Code for Information Interchange”.

It is a 7-bit code. So, each character is assigned a 7-bit binary value. This value is used by the digital systems (e.g. Computers) to correctly store and manipulate text data.

Originally used on non IBM systems. Basis of most currently used digital systems. Need to add an additional bit to make the code 8-bit

to be stored in a byte. Two options are provided: Simply add 0 in the MSB. Use it in even or odd-parity error detection coding as will be

seen later.

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ASCII Table

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Character Coding -- EBCDIC

EBCDIC stands for “Extended Binary Coded Decimal Interchange Code”.

8-bit code.

Originally used in IBM mainframes.

Provide a larger set of codes than ASCII.

Less common than ASCII.

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Character Coding -- Unicode

Designated as the universal code for text coding.

Comes to provide a large set of codes, i.e. solve the limitations found in ASCII and EBCDIC.

16-bit code.

Used in the Internet.

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Error Detection Codes I

Data communication requires special codes and algorithms to determine whether the received data is the same as the one transmitted, i.e. error free.

The simplest error detection code is based on adding a parity bit for ASCII code.

Parity means that two things are equal (identical). Simply add a 1 or 0 to the MSB in ASCII (parity) to

make the number of ones in that byte either odd (odd parity) or even (even parity).

Even parity is most common. E.g.: for 0111100 add 0 to MSB to obtain an even

parity and 1 to obtain an odd parity.

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Error Detection Codes II

The sender and the receiver know in advance the used parity type (odd or even).

The sender add the parity bit to each byte before transmission.

The receiver checks the parity before decoding the data. Assume that even parity is used, if the number of ones in a byte found to be odd, this means that there is an error in this byte.

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Error Detection Codes III

Disadvantages: Both even and odd parity can detect odd number

of error bits only. So, if two bits are in error for example, the parity remains correct.

Parity cannot determine the locations of the bits in error.

Cannot determine the number of bits in error.

What to do when detecting an error in the received message? Ask the sender to retransmit the message. Or use an error correction algorithm to recover the

original message.

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Gray Code

Also, known as “The Reflected Binary Code”. The main idea of Gray code is that any two

subsequent codes differ in one bit only. Useful for error detection and correction in

digital communication systems. Manly used in applications that produce

values in an increasing (or decreasing) order, i.e. similar to the counter concept.

Gray code can be 1-bit, 2-bit, 3-bit, etc., based on the number of values needed to be coded.

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3-bit Gray Code

Decimal Binary Gray Code

0 000 000

1 001 001

2 010 011

3 011 010

4 100 110

5 101 111

6 110 101

7 111 100

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Gray Code Conversion I

The first method includes conversion in one direction only (from binary value to Gray).

Start from the LSB, if the next bit (in the higher significance location) is 1 invert the current bit. If it is 0 then do not change the current bit.

E.g. convert 11011 to Gray code.

11011

1 0 1 1 0

Answer = 10110 Inverted (10)

since the bit after it is 1

Remains the same since the bit after it is 0

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Gray Code Conversion II

The second method includes conversion in two directions and it is based on the usage of the XOR logical function.

Convert Binary to Gray as follows: Copy the most significant bit For each smaller i do G[i] = XOR(B[i+1],

B[i]) (to convert binary to Gray)

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Gray Code Conversion III

Convert Gray to Binary as follows:

Copy the most significant bit

B[i] = XOR(B[i+1], G[i]) (to convert Gray to binary)

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Decimal Coding I

BCD is the main example of decimal coding. BCD Code (Binary Coded Decimal): A code used to

represent each decimal digit of a number by a 4-Bit Binary Value.

Valid Digits for 0-9 are (0000 to 1001) the binary codes 1010 to 1111 are invalid.

So, the BCD code of each digit is the binary conversion of that decimal digit.

BCD comes to provide: A solution for the difficulty of the conversion between binary

and decimal. Easier to be understood by humans.

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Decimal Coding II

Decimal BCD

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

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Decimal Coding III

For a compound number of decimal (multi-digit number) the BCD of that number is simply the substitution of the BCD code of each digit.

E.g. 12310 000100100011 (BCD)

So, n digit decimal number needs 4*n binary bits to be represented in BCD.

Pay attention to number justification when it is in BCD. Also, note that numbers in BCD are decimal not binary numbers.

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Decimal Coding IV

Note that a BCD digit needs 4 bit to be stored. The smallest storage unit in computer is byte

which is larger than what needed by 1 BCD digit.

There are two types of BCD code depending on how to store BDC digits in a byte.

BCD code types: Packed BCD: a byte can store two BCD digits. So, for

example, the number 867810 needs 2 bytes. Unpacked BCD: a byte can store one BCD digit in the

lower nibble while the higher nibble contains zeros. So, for example, the number 867810 needs 4 bytes.

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Additional Notes

This lecture covers the following material from the textbook:

Chapter 1: Section 1.7