# Why should a Reed Solomon code be a 'narrow sense' BCH -code?

I am preparing a presentation about the Reed Solomon code for a course I follow at uni. However, I have a question:

$\textbf{Why should the Reed Solomon code be narrow sense?}$

What I understand is: consider a BCH code with design parameters $b$ and design distance $d_{\text{BCH}}$ of length $n$ over a field $\mathbb{F}_q$ of cardinality $q$, with $q = p^k$ for some prime $p$. In order to find a generator polynomial of the BCH code, we need to factorize $x^n -1$ over this field $\mathbb{F}_q$. Let $\beta$ be a primitive $n$th root of unity, then I will denote the minimal polynomial of $\beta^i$, with $i \in \{0, 1, \ldots, n-1\}$ by $m^{(i)}(x)$. As a result, the generator polynomial of the given BCH code would be $$\text{lcm}(m^{(b)}(x), \ldots, m^{(b + d_{\text{BCH}} - 2)}(x)).$$ (I learned this in a course where we saw the BCH code).

What I understand from the text I read (being 'Coding Theory: a first course' by Henk C.A. van Tilborg) is that the Reed Solomon code is a BCH code where $b = 1$ (indicated as 'narrow sense') and $n = q -1$, excluding binary codes. Now I know that $x^{q-1} -1$ factors into linear factors over $\mathbb{F}_q$, showing that the generator polynomial has degree $d_{\text{BCH}} - 1$. From this and the fact that cyclic codes satisfy $$k \leq n - d +1$$ where $k$ is the dimension of the cyclic code (so the cardinality of the code is $q^k$), $d$ is the minimal distance of the code, we find that $$d_{\text{BCH}} = d.$$

$\textbf{ACTUAL QUESTION}$: However, all of this does not require that the code is narrow sense... Is there anyone who knows why this should hold? Or is it just some convention?

$\textbf{REMARK}$: The course notes I used (Coding theory, a first course) as additional lecture about Reed Solomon, does not use $b$, but the 'defining set of a cyclic code $I$'. This is the union of the cyclotomic cosets of corresponding to the minimal polynomials used in the generator polynomial of the BCH code. However, I figured out that these two should correspond (the $b$ was introduced in the course I followed).

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 $\beta$) was that it led to the fewest number of transistors in the decoder implementation in the hardware technology of the 1970s.
However, the choice $b=1$ (which gives a narrow-sense BCH code) is convenient for the purposes of exposition and in connecting together different descriptions of Reed-Solomon codes. See, for example, this answer of mine which shows that a code obtained via the original definition of Reed-Solomon codes is in fact a narrow-sense BCH code.
• This explains a lot! I really appreciate your answer and I will accept as soon as possible. For now, +1 sir! However, could you explain what the $[255, 223]$ stands for? I know that 225 is because of $q -1$, so its the length of the code, but what is the 223? – Student Mar 6 '17 at 16:43
• @Student An $[n,k]$ code over $\mathbb F_q$ is a $k$-dimensional subspace of the $n$-dimensional vector space $\mathbb F_q^n$. This is standard notation and I am surprised that you have not encountered it before. – Dilip Sarwate Mar 6 '17 at 17:03