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Chapter 10

Error Detection

and

Correction

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The Hamming distance between two

words is the number of differences

between corresponding bits.

Note

The minimum Hamming distance is the

smallest Hamming distance between

all possible pairs in a set of words.

Note

Find the minimum Hamming distance of the coding

scheme in Table 10.1.

Solution

We first find all Hamming distances.

Example 10.

The d

min

in this case is 2.

To guarantee the detection of up to s

errors in all cases, the minimum

Hamming distance in a block

code must be d

min

= s + 1.

Note

The minimum Hamming distance for our first code

scheme (Table 10.1) is 2. This code guarantees detection of

only a single error. For example, if the third codeword

(101) is sent and one error occurs, the received codeword

does not match any valid codeword. If two errors occur,

however, the received codeword may match a valid

codeword and the errors are not detected.

Example 10.

Figure 10.8 Geometric concept for finding d

min

in error detection

Figure 10.9 Geometric concept for finding d min

in error correction

A code scheme has a Hamming distance d

min

= 4. What is

the error detection and correction capability of this

scheme?

Solution

This code guarantees the detection of up to three errors

(s = 3), but it can correct up to one error. In other words,

if this code is used for error correction, part of its capability

is wasted. Error correction codes need to have an odd

minimum distance (3, 5, 7,... ).

Example 10.

103 LINEAR BLOCK CODES

103 LINEAR BLOCK CODES

Almost all block codes used today belong to a subset

Almost all block codes used today belong to a subset

called

called linear block codes

linear block codes

. A linear block code is a code . A linear block code is a code

in which the exclusive OR (addition modulo-2) of two

in which the exclusive OR (addition modulo-2) of two

valid codewords creates another valid codeword.

valid codewords creates another valid codeword.

Minimum Distance for Linear Block Codes

Some Linear Block Codes

Topics discussed in this section:

Topics discussed in this section:

Let us see if the two codes we defined in Table 10.1 and

Table 10.2 belong to the class of linear block codes.

1. The scheme in Table 10.1 is a linear block code

because the result of XORing any codeword with any

other codeword is a valid codeword. For example, the

XORing of the second and third codewords creates the

fourth one.

2. The scheme in Table 10.2 is also a linear block code.

We can create all four codewords by XORing two

other codewords.

Example 10.

In our first code (Table 10.1), the numbers of 1s in the

nonzero codewords are 2, 2, and 2. So the minimum

Hamming distance is d

min

= 2. In our second code (Table

10.2), the numbers of 1s in the nonzero codewords are 3,

3, and 4. So in this code we have d

min

Example 10.

Table 10.3 Simple parity-check code C(5, 4)

Figure 10.10 Encoder and decoder for simple parity-check code