Question

The following message was originally transmitted with odd parity in each short bit string. In which strings have errors definitely occurred?

Short answer

Errors occurred in strings with an even count of 1s.

Step-by-step solution

Understanding Odd Parity

Odd parity means that each bit string should have an odd number of 1s. We'll check each string to ensure it has an odd count of 1s.

Analyzing Each String

For each short bit string provided: 1. Count the number of 1s. 2. Determine if the count is odd or even. 3. Identify if there is an error based on the parity requirement.

Example Verification

Let's say one of the strings is 1101010. Count the 1s: there are 4. Since 4 is even, and we are expecting an odd number of 1s, an error must have occurred in this string.

Repeat for All Strings

Perform the counting and parity check for each bit string. Any string with an even number of 1s will be identified as having had an error.

Key concepts

Odd Parity

Odd parity is a method used in data transmission to ensure each segment of transmitted data contains an odd number of '1's. This helps to identify errors when data is read on the receiver's end. When the number of '1's in a bit string matches the requirement (odd), the data is considered accurate. If the number of '1's is even, it indicates that an error has occurred, possibly due to interference or a signal glitch during transmission.

Parity checking does not detect all kinds of errors but is particularly useful for detecting errors where a single bit has flipped during transmission. While other methods exist for error detection, odd parity offers a simple and effective first line of defense. By maintaining an odd number of '1's, receivers can quickly flag a string with an even count as containing an error.

Binary Strings

Binary strings are sequences of '0's and '1's that encode information in a format understandable by digital systems. Each position in a binary string represents a power of two, starting from the rightmost digit. These strings can vary in length and complexity, depending on the data they represent.

In the context of parity error checking, binary strings are carefully constructed to follow specific rules (like odd parity). Ensuring that binary strings follow these rules allows for efficient communication between digital devices. Odd parity in binary strings is applied by potentially adjusting the final bit so that the total count of '1's in the string is odd, ensuring data integrity throughout the process.

Error Detection

Error detection refers to techniques used to identify and correct errors in data during transmission or storage. Parity checking is a fundamental form of error detection, especially useful when the data involved is relatively small or simple.

Using odd parity, error detection is simplified to counting the '1's in each binary string. If a string does not have an odd count of '1's, an error is presumed, prompting further investigation or retransmission of the data. This technique is quick and requires minimal computational resources, making it ideal for real-time data communication systems where speed is crucial.

Bit Counting

Bit counting is the process of counting how many '1's are present in a binary string. This is a straightforward way to verify adherence to parity checks like odd parity. For each string, the '1's are tallied, and this count is continuously evaluated against the expected parity rule.

The simplicity of bit counting allows it to be employed in various systems, where ensuring data accuracy is critical but the systems must also run efficiently. Bit counting is integrated into many hardware and software solutions for error checking, and when combined with parity checks, it helps maintain data integrity across multiple computing environments.

• Counting helps identify errors in data communication by quickly pinpointing incorrect strings.
• Many digital systems perform bit counting automatically during data processing.
• Hence, bit counting is an essential part of error detection and correction mechanisms.