Practical Session 11

Codes

Hamming Distance

• General case: The distance between two code words is the amount of 1-bit changes required to reach from one word to the other.

• Binary case: Number of 1’s in the XOR between the words.

• Hamming Distance is the minimum distance between the code words.

Hamming Distance

• Intuition (code graph):• Example: d=2

00

10 11

01

Error Detection

• Hamming Distance: d.• Can Detect d-1 errors.

Fixing Erasures

• Can fix d-1 erasures.

Error Correction

• Can fix Errors2

1d

Example

• Given four “data” words, can we use 5-bit code words for fixing 1 error?

• Answer: Yes. We need a hamming distance of 3 to fix 1 error. We can create code words with independent graph “spheres”.

Parity Check

• The most basic error check.• Used to test one character.• Appended to the end of the message.• Can detect 1 error.• Example (Even parity):

10010100 1

10010000 1

Block Character Checksum

• Uses vertical and horizontal parity bits to detect double errors.

• Used for transmitting a block of characters.• BCC is used to detect two errors.• Odd parity calculated for each row.• Even parity calculated for each column.

BCC - Example

• Two errors in the character ‘N’.• Parity doesn’t change.• BCC changed.

Cyclic Redundancy Check

• A CRC is an error-detecting code• An n-bit CRC, applied to a data block of

arbitrary length, will detect any single error burst not longer than n bits

• Will detect a fraction 1-2-n of all longer error bursts

• The simplest error-detection system, the parity bit, is in fact a trivial CRC: it uses the two-bit-long divisor 11.

Calculation Steps - Sending

1. A generator is chosen.– This is a sequence of bits, of which the first and

last are 1.– This sequence is used with the bits of the

message to calculate a check sequence which has 1 fewer bits than the generator.

2. The check sequence is appended to the original message

Calculation Steps - Receiving

• At the receiver, the same calculation is performed on the message and check sequence combined.

• If the result is 0 no transmission error is assumed to have occurred.

Example

• Compute 8-bit CRC for an 8-bit message: ‘W’.• ASCII for ‘W’: 8710 = 5716

• Select Generator: CRC-8-ATM– Polynomial: x8 + x2 + x + 1– Corresponds to: 100000111

• Two different orders are possible for the message.– Msbit-first: 010101112 = x6 + x4 + x2 + x + 1

– Lsbit-first: 111010102 = x7 + x6 + x5 + x3 + x

Example – Generating CRC Code

• We will focus on Msbit-first order.• Append 8 bits to message:• 0101011100000000• Perform long-division of message by

generator.– Quotient is not needed.– Instead of subtraction perform simple XOR.– CRC code is the remainder.

Example – Generating CRC Code

0101011100000000 100000111

CRC Code: 10100010

100000111

001011011000000

100000111

0011010110000

100000111

01010101100

100000111

0010100010

xor

xor

xor

xor

Example – Checking Message

• Similar to generating the CRC code.• Append CRC code to the message (instead of

0’s): 0101011110100010• Perform long division by the generator.• If the reminder is not 0: An error occurred.

Example – Checking Message

0101011110100010 100000111

Result: 00000000

CRC Polynomials (Common)

CRC-8-ATMx8 + x2 + x + 1

CRC-16-IBMx16 + x15 + x2 + 1

CRC-16-CCITTx16 + x12 + x5 + 1

CRC-32-IEEE 802.3x32 + x26 + x23 + x22 + x16 + x12 + x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1