Is it Possible to Create a 3-bit Code with 4 Codewords for Single Error Detection?

Introduction

In coding theory, error detection is a critical component in ensuring the reliability of data transmission. This article explores the possibility of creating a 3-bit code with 4 codewords that can detect any single error. We will delve into the concepts of codewords, the calculation of Hamming distance, and the feasibility of such a code.

Codeword Basics

A binary string of length 3 can represent 8 possible combinations. These are the possible codewords within a 3-bit system. Each codeword can be represented as a combination of 0s and 1s, such as 000, 001, 010, and so on.

Single Error Detection

A code can detect a single error if, for any two distinct codewords, the Hamming distance between them is at least 3. The Hamming distance is the number of positions at which the corresponding bits are different. For example, the Hamming distance between 000 and 111 is 3, and between 000 and 001 is 1. To detect a single error, this distance must be maintained so that flipping a single bit in a codeword does not result in another valid codeword.

Feasibility Analysis

The feasibility of creating such a code depends on the properties of the Hamming distance and the number of codewords.

Maximum Codewords

To detect single errors, the maximum number of codewords (M) that can be used is given by the formula:

M ≤ 2^{n-1}

Given (n 3) (the length of the codewords) and needing at least a Hamming distance of 2 for single error detection:

M ≤ 2^{3-1} 2^2 4

This means we can have at most 4 codewords, but achieving this with a 3-bit system is challenging due to the limited number of possible codewords.

Choosing Codewords

Let's consider some examples and the distances between them:

000 111 001 110

Calculating the Hamming distance between these codewords:

Distance 000 and 111 is 3 Distance 000 and 001 is 1 Distance 000 and 110 is 2 Distance 111 and 001 is 2 Distance 111 and 110 is 1 Distance 001 and 110 is 2

From this analysis, we see that the pairs 000 and 111 have a distance of 3, but the distances between the other codewords do not all meet the minimum distance requirement of 2. Therefore, these codewords cannot be used together to form a code that detects all single errors.

Conclusion

After further consideration, it turns out that it is not possible to create 4 codewords of length 3 that can detect any single error. The maximum number of codewords that can be used while ensuring single error detection with a Hamming distance of at least 2 is actually not achievable with 4 distinct codewords in a 3-bit binary string.

If you want to create a code that can detect single errors, you can have up to 3 codewords. An example of such a code is:

000 111 101

This set of codewords satisfies the single error detection condition.