1. Reed-Solomon Parity in $GF(2^8)$
QR codes implement non-binary Reed-Solomon error correction over the finite field Galois Field 256 ($GF(2^8)$). In this algebra, numbers from 0 to 255 represent 8-bit bytes, and mathematical operations (addition, subtraction, multiplication, and division) are performed modulo an irreducible primitive polynomial:
$$p(x) = x^8 + x^4 + x^3 + x^2 + 1$$
When generating a QR code, the raw data codewords are treated as coefficients of a polynomial $D(x)$. This polynomial is multiplied by $x^{2t}$ and divided by a generator polynomial $g(x)$:
$$g(x) = (x - alpha^0)(x - alpha^1)(x - alpha^2)dots(x - alpha^{2t-1})$$
The remainder polynomial $R(x)$ contains the parity codewords that are appended to the payload.
2. Comparison of the Four Standard Levels
| Level | Identifier | Recovery Headroom | Recommended Use Case |
|---|---|---|---|
|
Level L | Low | ~7% of codewords | High-density data, clean digital screens, micro-labels |
|
Level M | Medium | ~15% of codewords | Default standard for flyers, brochures, and web links |
|
Level Q | Quartile | ~25% of codewords | Industrial environments, logistics, outdoor posters |
|
Level H | High | ~30% of codewords | Embedded logo designs, harsh factory floors, vehicle decals |
3. The Logo Overlay Trade-off
When adding a brand logo to the center of a QR code, you are deliberately obliterating valid data modules. If you place a logo covering 20% of the symbol area:
* A code generated at
Level L (7%) or
Level M (15%) will become completely unreadable.
* A code generated at
Level H (30%) will still retain approximately 10% of recovery margin for physical scratches and optical noise.
Always configure your QR generator to Level H before embedding custom center artwork.
Putting this into practice
Start with the smallest real test you can run. Generate the code, print it at the exact size you plan to use, and scan it with two different phones in the lighting where it will live. Check that the destination page loads quickly on a mobile connection and that the call to action next to the code tells people what they will get. If anything feels slow or unclear, fix it before you print a full batch, because reprints cost more than a short test. Keep a note of the size, material, and placement that worked so the next batch starts from a proven setup.
Common mistakes to avoid
The most common mistakes are easy to avoid once you know them. Long payloads make modules tiny, low contrast makes edges hard to read, and text placed too close to the border eats the quiet zone the scanner needs. Covering a corner finder pattern with a logo or a sticker breaks detection entirely, and a code placed behind glare or on a curved surface fails even when the print is perfect. Review each of these points on a proof copy, and treat any failure as a reason to adjust the design rather than a reason to blame the scanner.
Quick checklist
- Keep the payload as short as practical so modules stay large.
- Keep dark modules on a light background with strong contrast.
- Leave a clear quiet zone on all four sides of the symbol.
- Test the printed code at the real viewing distance and lighting.
- Confirm the destination page loads fast on a phone.
- Recheck the code after any change to the link, design, or placement.
Related tools and guides
- Use the [QR size calculator](/qr-size-calculator) to match the printed size to the expected scan distance before you commit to a print run.
- The [error correction simulator](/error-correction-simulator) shows how damage tolerance changes the density of the symbol at each level.
- Test every code with the [QR scanner](/qr-scanner) on more than one phone before you publish or print it.
- Create a fresh code with the [URL QR code generator](/url-qr-code-generator) when the destination is a web page.
- Run unfamiliar destinations through the [QR safety checker](/qr-safety-checker) before you share a code widely.
- The [blog](/blog) collects more guides on sizing, printing, security, and payments for QR codes.
