Error-detection capability of single-parity schemes: Can simple parity checking detect only an odd number of bit errors in a codeword (and miss even numbers of bit flips)?

Introduction / Context:
Parity bits are the simplest error-detection mechanism used in memory systems, serial links, and legacy protocols. A single parity bit forces the total number of 1s in a codeword to be either even (even parity) or odd (odd parity). Understanding exactly what error patterns parity detects—and which it cannot—guides the selection of stronger codes when needed, such as checksums, CRCs, or Hamming codes.

Given Data / Assumptions:

• Single parity bit appended to a data word.
• Even or odd parity convention is known to both transmitter and receiver.
• Independent bit-flip error model (no burst-correction mechanisms assumed).

Concept / Approach:
Let w be the number of 1s in the transmitted data. The parity bit is set so that w + p has a fixed parity (even or odd). If an odd number of bits flip during transmission, the parity of the received word toggles, producing a mismatch flag. If an even number of bits flip, parity returns to the original parity class and the simple parity check fails to detect the error. Thus, parity detects any odd-weight error pattern (1, 3, 5, … flips) and misses all even-weight error patterns (2, 4, 6, … flips). The minimum Hamming distance of a single-parity code is 2, enabling detection of all single-bit errors but not correction and not detection of double-bit errors.

Step-by-Step Solution:

Verification / Alternative check:
Compute Hamming distance: with one parity bit, any valid codeword differs from any other by at least 2, confirming detect-1 capability and failure to detect all double-bit errors.

Why Other Options Are Wrong:
Odd vs even parity choice does not affect the odd-error detection rule; both detect odd numbers of flips. Hamming distance > 2 requires more redundancy than single parity. Applicability is general to binary data, not restricted to ASCII.

Common Pitfalls:
Assuming parity can correct errors (it cannot); assuming it detects double-bit errors (it does not); mixing parity with CRC behavior, which has different detection guarantees.

Final Answer:
Correct