Question

When i started reading about Qr Codes every article browsed i can see one QR Code Exponents of αx Table which having values for the specific powers of αx . I am not sure how this table is getting created. Can anybody explain me the logic behind this table.

For reference the table can be found at http://www.matchadesign.com/_blog/Matcha_Design_Blog/post/QR_Code_Demystified_-_Part_4/#

Answer 1

(The zxing source code for this might help you.)

It would take a lot to explain all the math here. For the Reed-Solomon error correction, you need a Galois field of 256 elements (nothing fancy — just a set of 256 things that have addition and exponentiation and such defined.)

This is defined not in terms of numbers, but in terms of polynomials whose coefficients are all 0 or 1. We work with polynomials with 8 coefficient — conveniently these map to 8-bit values. While it’s tempting to think of those values as numbers, they’re really something different.

In fact for addition and such to make sense such that all the operations land you back in a value in the Galois field, all the results are computed modulo an irreducible polynomial in the field. (Skip what that means now.)

To make operations faster, it helps to pre-compute what the powers of the polynomial “x” are in the field. This is alpha. You can think of this as “2”, since the polynomial “x” is 00000010, though that’s not entirely accurate.

So then you just compute the powers of x in the field. Because it’s a field you’ll hit every element of the field this way. The sequence seems to be the powers of two, which it happens to map to for a short while, until the first “modulo” of the primitive polynomial takes effect. Multiplying by x is indeed still something like multiplying by 2 but it’s a bit of coincidence in this field, really.