Converting between number code systems — binary, BCD, Gray code, Excess-3 — is a recurring category of digital design problem, and every conversion follows the same pattern: each output bit is its own independent Boolean function of the input bits, minimized separately. BCD-to-Gray is covered in full in the worked example; this covers BCD-to-Excess-3 as a second, equally verified case.
What Excess-3 code is
Excess-3 represents a decimal digit as its binary value plus 3 — digit 0 becomes binary 3 (0011), digit 9 becomes binary 12 (1100). It was historically used because it makes certain arithmetic operations (like nines' complement) simpler in BCD arithmetic circuits — the "plus 3" offset isn't arbitrary, it shifts the valid digit range so useful symmetries appear in the resulting bit patterns.
| Digit | BCD | Excess-3 |
|---|---|---|
| 0 | 0000 | 0011 |
| 1 | 0001 | 0100 |
| 2 | 0010 | 0101 |
| 3 | 0011 | 0110 |
| 4 | 0100 | 0111 |
| 5 | 0101 | 1000 |
| 6 | 0110 | 1001 |
| 7 | 0111 | 1010 |
| 8 | 1000 | 1011 |
| 9 | 1001 | 1100 |
All four output bits, minimized and verified
E0 stands out — a single literal. The least significant Excess-3 bit is simply the complement of the least significant BCD bit, for every valid digit, no exceptions. That's a direct, verifiable consequence of adding an odd number (3) to any value: it always flips the least significant bit. Worth noticing on sight for any "add an odd constant" code conversion, not just this one.
Same technique, different output table
Whether the target is Gray code, Excess-3, or another code conversion, build a truth table for each output bit and apply don't-cares only to input codes that are guaranteed never to occur (see Handling Don't-Care Conditions in K-Maps). Minimize each bit to create a baseline, then compare the resulting terms across output bits for possible sharing. That circuit-level step is covered in multi-output Boolean minimization.
Try a code conversion of your own directly on the Truth Table to K-Map converter.