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Just a quick question. What Galois Field is best if you need a minimum Hamming distance of $n$? I want to design a BCH generator polynomial that is acccording to this Hamming distance.

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This reference provides specific implementations for three error correcting BCH generators with a minimum hamming distance of 7

BCH codes

Note, also refer to Peterson's Table of Irreducible Polynomials which lists all primitive and irreducible polynomials over GF(2):

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  • $\begingroup$ Thank you! What if my message is 16 bits, can I use a GF (2^6) for a k = 16 bits even though my dmin is still at 7? $\endgroup$ – LeBlanc Lord May 9 '17 at 13:59
  • $\begingroup$ @LeBlancLord See the first page of the first link which outlines the block length n, the number of parity bits and minimum distance for a t-error corecting BCH code. In your case t=3, the minimum distance is >=7 and the number of parity-check digits is <= mt where the block length = $2^m-1$. With t small, n-k is exactly mt. So solving this for t=3 and k=16 results in m=5. So if I did that correctly you would have 16 message bits, and 3m=15 parity bits for a block length of 31, and dmin is 2t+1=7. Do you agree? $\endgroup$ – Dan Boschen May 10 '17 at 9:43

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