Parity bits and 3-bit even parity generator

Importance of parity bits

A parity bit is an extra bit added to a data word so that the total number of 1s is even (even parity) or odd (odd parity). Its importance in communication:

• Error detection: noise can flip a bit during transmission. The receiver recounts the 1s; a wrong parity shows that an error occurred.
• Simple and cheap: only one extra bit and a few XOR gates are needed at each end.
• Detects all single-bit (odd-number) errors, which are the most common in many links. (It cannot detect two-bit errors and cannot correct errors.)
• Used in serial links (UART), memory (parity RAM) and as the base of Hamming codes.

3-bit even parity generator

Inputs: data bits $A, B, C$. Output: parity bit $P$ chosen so that $A, B, C, P$ together have an even number of 1s.

$$P = \Sigma m(1,2,4,7)$$

The K-map is a checkerboard (no adjacent 1s), so it is simplified with XOR:

$$\begin{aligned} P &= A'B'C + A'BC' + AB'C' + ABC \\ &= A'(B \oplus C) + A(B \odot C) \\ &= A \oplus B \oplus C \end{aligned}$$

 A --+
     +--[XOR]--+
 B --+         +--[XOR]--- P
 C ------------+

Working: $P$ is 1 when the data has an odd number of 1s, which makes the total even. Example: data $101$ has two 1s, so $P = 1 \oplus 0 \oplus 1 = 0$, and the transmitted word $1010$ has two 1s (even). At the receiver, a 4-bit checker $PEC = A \oplus B \oplus C \oplus P$ gives 0 for no error and 1 if a single bit has flipped.