Question

Suppose a 24-bit code is generated by representing each \(s y m b o l\) by three consecutive copies of its ASCII representation (for example, the symbol \(A\) is represented by the bit string 010000010100000101000001 ). What error- correcting properties does this new code have?

Short answer

The code can correct any single segment error within the 24 bits by using majority rules among the three segments.

Step-by-step solution

Understanding the Problem

We are asked to analyze a code that generates a 24-bit string by repeating each symbol's 8-bit ASCII code three times consecutively. For example, the letter 'A' with ASCII code 01000001 is represented as 010000010100000101000001. We need to find out the error-correcting properties of this coding scheme.

Analyzing Redundancy

The code is designed so that each 8-bit ASCII symbol is repeated three times in the 24-bit block. This creates redundancy, allowing us to detect and correct certain types of errors by comparing the three segments.

Error Detection Capability

Due to the triplication of the ASCII code, if one of the three segments is incorrect, it can be detected since it won't match the other two. Hence, any single-bit error in one block can be detected.

Error Correction Capability

For error correction, if at most one of the three 8-bit segments is corrupted, then the two matching segments will be assumed as correct. This allows for correcting up to one erroneous segment within any 24-bit codeword.

Limitations of the Code

This coding scheme cannot correct a majority of bit errors that occur in more than one of the three segments, as there would be no 'majority' to establish the correct value. Additionally, if exactly one bit is corrupted in each of the three segments, the corruption won't be detected.

Conclusion on Error-Correcting Properties

This code can detect and correct any single segment error (out of three) but is vulnerable to errors affecting multiple segments. Thus, it offers simple error correction for low error rates.

Key concepts

ASCII representation

The ASCII representation is a character encoding standard that stands for American Standard Code for Information Interchange.
Each character, whether a letter, number, or symbol, is represented by a unique 8-bit binary number.
For example, the letter 'A' is represented as 01000001 in binary.

• ASCII provides a unique code to each character, which helps in digital communication and storage.
• Being limited to 8 bits means that ASCII can represent up to 256 different characters.
• It is a foundational scheme used in computers and various devices for representing text.

For error-correcting codes, ASCII acts as the base encoding of each symbol.
This encoding is then manipulated, such as by duplication, to enhance properties like error detection and correction.

redundancy in coding

Redundancy in coding refers to adding extra bits to the encoded data.
This allows the system to have multiple representations and increases the capacity to detect or correct errors.

• In the given problem, each ASCII character is repeated three times, creating redundancy.
• This tripling forms a 24-bit code from the original 8-bit ASCII.
• Redundant information helps in comparing parts of the data to verify accuracy.

Though redundancy increases the size of data being stored or transmitted, it crucially enables the detection and correction of errors.
By comparing the repeated segments, it is possible to identify discrepancies that might suggest a mistake during data transmission or storage.

error detection

Error detection is the process of identifying errors in data transmissions.
With the redundancy introduced, every triplicated 8-bit segment can be checked against the others.

• If one segment differs from the others, an error is detected.
• This scheme allows detection of any single-bit error in one of the three sequences.
• Since two segments will match, they are likely correct, isolating the erroneous third.

This method, known as a "majority vote" system, provides a simple yet effective way to identify corrupted data.
However, it only works efficiently under conditions where errors are rare, affecting a single segment only.

error correction capability

The error correction capability of this coding is limited to fixing errors in one of the three 8-bit segments.
The assumption is that the correct data appears in the majority of segments.

• When two segments match and one differs, the differing segment is assumed to be incorrect.
• This allows the code to correct an error affecting a single segment.
• However, if errors affect multiple segments, the code loses its ability to correctly fix the errors.

This simple error correction works under the assumption of low error rates.
It is vulnerable if multiple matching errors occur or if errors are dispersed equally across all segments.
Thus, while useful, this method may not suffice for more error-prone environments where data integrity is crucial.