Gray Code Problem: Analysis

The question asks us to identify the correct Boolean function that maps the 4-bit Gray code representation of a number n, denoted as $h_3 h_2 h_1 h_0$, to a specific bit ($g_2$) of the Gray code representation of the next number $(n + 1) \pmod{16}$. We need to find which of the given functions accurately represents $g_2$. The notation $\Sigma$ represents the sum of minterms for which the function evaluates to 1, based on the decimal value of the input Gray code $h_3 h_2 h_1 h_0$.

Gray Code Generation: n and n+1

To solve this, we first list the standard 4-bit Binary-Reflected Gray Code (BRGC) sequence. We construct a truth table that shows the Gray code for n ($H = h_3h_2h_1h_0$), its corresponding decimal value (which serves as the minterm index), the next number $(n+1) \pmod{16}$, and the Gray code for that next number ($G = g_3g_2g_1g_0$). We specifically track the value of the $g_2$ bit.

Function Derivation: g₂

By examining the truth table, we identify the rows where the output bit $g_2$ is equal to 1. The function $g_2$ should evaluate to 1 for the input Gray codes $H$ corresponding to the minterms (decimal values of $H$) where $g_2$ is 1.

From the table, $g_2 = 1$ when the input Gray code's decimal value is 2, 4, 5, 6, 7, 12, 13, or 15.

Therefore, the function $g_2$ expressed as a sum of minterms is:

Result Verification: Function Comparison

Comparing this derived function $g_2 = \Sigma(2, 4, 5, 6, 7, 12, 13, 15)$ with the provided options, we find that it exactly matches the function stated in Option 3.