3-bit even parity generator and 4-bit checker

Parity is used to detect single-bit errors. With even parity, the transmitted word (data + parity bit) always has an even number of 1s.

3-bit even parity generator

Inputs $A, B, C$; output $P$ makes the total number of 1s even.

$$\begin{aligned} P &= \Sigma m(1,2,4,7) = A'B'C + A'BC' + AB'C' + ABC \\ &= A'(B \oplus C) + A(B \odot C) = A \oplus B \oplus C \end{aligned}$$

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

4-bit even parity checker

The receiver gets $A, B, C, P$ and produces the parity error check $PEC$: 0 if the count of 1s is even (no error), 1 if odd (error).

$PEC = 1$ for minterms 1, 2, 4, 7, 8, 11, 13, 14 (odd number of 1s). The K-map is a checkerboard, so

$$PEC = A \oplus B \oplus C \oplus P = (A \oplus B) \oplus (C \oplus P)$$

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

Example

Data $110$: generator gives $P = 1 \oplus 1 \oplus 0 = 0$; sent word $1100$. If received correctly, $PEC = 1 \oplus 1 \oplus 0 \oplus 0 = 0$ (no error). If bit C flips (received $1110$), $PEC = 1$, so an error is flagged.