On Reed Solomon Codes

Question

I am trying to determine all possible values (parameters $n,k$) for which an RS-code exists over $GF(2^9)$. Using definition of RS-code, we know that $n|q-1$ and the designed distance $\delta \ge 2$ with $k=n+1- \delta$. The divisors of $511$ include $7,73$ and $511$. When $n=7$, the values are $1\le k \le 6$ for $2 \le \delta \le 7$. To be clear RS-Code $(7,6),(7,5),(7,4),(7,3),(7,2),(7,1)$ exists with distance $2,3,4,5,6,7$ respectively and the error correcting capabilities are $0,1,1,2,2,3$ respectively. The same procedure is performed for other divisors of $511$ which are $73$ and $511$. Does this look okay? or am I missing something? Is there an efficient/faster way to do this?

Answer 1

Your argument is correct. And since RS codes are MDS and obey $d_k=n-k+1,$ the parameter choices are as you state, and you obtain distance $d$ by choosing $d-1$ consecutive powers of an element $\beta$ of order $n,$ as roots of the generating polynomial. The choice of where to start the string of consecutive powers may matter in terms of efficiency of implementation, as described in the answer to the question here :

It is not necessary for a Reed-Solomon code to be a narrow-sense BCH code. Indeed, the [255,223] "NASA standard" cyclic Reed-Solomon code over $\mathbb{F}_{2^8}$ that was widely used many years ago (and still survives today in various other standards) is not a narrow-sense BCH code. One reason for that particular choice of b (and the choice of minimal polynomial of degree 8 whose root was β) was that it led to the fewest number of transistors in the decoder implementation in the hardware technology of the 1970s.