QR codes are 2-dimensional bar codes that encode arbitrary text strings. A common use of QR codes is to encode URLs so that people can scan a QR code (for example, on an advertising poster, building roof, volleyball bikini, belt buckle, or airplane banner) to load a web site on a cell phone instead of having to “type” in a URL.
QR codes are encoded using Reed-Solomon error-correcting codes, so that a QR scanner does not have to see every pixel correctly in order to decode the content. The error correction makes it possible to introduce a few errors (fewer than the maximum that the algorithm can fix) in order to make an image. For example, in 2008, Duncan Robertson took a QR code for “http://bbc.co.uk/programmes” (left) and introduced errors in the form of a BBC logo (right):
That's a neat trick and a pretty logo, but it's uninteresting from a technical standpoint. Although the BBC logo pixels look like QR code pixels, they are not contribuing to the QR code. The QR reader can't tell much difference between the BBC logo and the Union Jack. There's just a bunch of noise in the middle either way.
Since the BBC QR logo appeared, there have been many imitators. Most just slap an obviously out-of-place logo in the middle of the code. This Disney poster is notable for being more in the spirit of the BBC code.
There's a different way to put pictures in QR codes. Instead of scribbling on redundant pieces and relying on error correction to preserve the meaning, we can engineer the encoded values to create the picture in a code with no inherent errors, like these:
This post explains the math behind making codes like these, which I call QArt codes. I have published the Go programs that generated these codes at code.google.com/p/rsc and created a web site for creating these codes.
For error correction, QR uses Reed-Solomon coding (like nearly everything else). For our purposes, Reed-Solomon coding has two important properties. First, it is what coding theorists call a systematic code: you can see the original message in the encoding. That is, the Reed-Solomon encoding of “hello” is “hello” followed by some error-correction bytes. Second, Reed-Solomon encoded messages can be XOR'ed: if we have two different Reed-Solomon encoded blocks b1 and b2 corresponding to messages m1 and m2, b1 ⊕ b2 is also a Reed-Solomon encoded block; it corresponds to the message m1 ⊕ m2. (Here, ⊕ means XOR.) If you are curious about why these two properties are true, see my earlier post, Finite Field Arithmetic and Reed-Solomon Coding.
A QR code has a distinctive frame that help both people and computers recognize them as QR codes. The details of the frame depend on the exact size of the code—bigger codes have room for more bits—but you know one when you see it: the outlined squares are the giveaway. Here are QR frames for a sampling of sizes:
The colored pixels are where the Reed-Solomon-encoded data bits go. Each code may have one or more Reed-Solomon blocks, depending on its size and the error correction level. The pictures show the bits from each block in a different color. The L encoding is the lowest amount of redundancy, about 20%. The other three encodings increase the redundancy, using 38%, 55%, and 65%.
(By the way, you can read the redundancy level from the top pixels in the two leftmost columns. If black=0 and white=1, then you can see that 00 is L, 01 is M, 10 is Q, and 11 is H. Thus, you can tell that the QR code on the T-shirt in this picture is encoded at the highest redundancy level, while this shirt uses the lowest level and therefore might take longer or be harder to scan.
As I mentioned above, the original message bits are included directly in the message's Reed-Solomon encoding. Thus, each bit in the original message corresponds to a pixel in the QR code. Those are the lighter pixels in the pictures above. The darker pixels are the error correction bits. The encoded bits are laid down in a vertical boustrophedon pattern in which each line is two columns wide, starting at the bottom right corner and ending on the left side: