How to achieve upper bound on the minimum distance of a BCH code?

Question

Let $[2^{m}-1, k ,d]_2$ be a BCH code, and let $g(x)\in GF[2](x)$ be its generating polynomial. Let $\alpha^{i_1}, \alpha^{i_2}, \alpha^{i_3},...,\alpha^{i_t}$ are the different roots of $g(x)$ (not necessarily all the roots) such that $i_1,i_2,i_3,...,i_t$ form an arithmetic progression, then $t+1\leq d \leq 2t-1$. Now, my question is how to achieve the upper bound on the $d$? Is there a way of construction through which I can get $d=2t-1$?

Answer 1

There are generalizations of the BCH bound which may be sometimes applied. I don't know of any constructive way that will work in general. See

C Roos, A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound, Journal of Combinatorial Theory, Series A, 33(2):1982

which is available here.

In particular the Roos bound implies the Hartmann-Tzeng bound which was the first improvement on the BCH, as indicated below. The set of zeroes defining the cyclic code is denoted by $N$ in this paper.

Proposition If $N=\{\alpha^{b+i_1 c_1+i_2 c_2}: 0\leq i_1\leq \delta-2, 0\leq i_2 \leq s\}, $ where $\delta\geq 2, \gcd(n,c_1)=1,$ and $\gcd(n,c_2)<\delta,$ then the minimum distance of the code with defining set $N$ is at least $\delta+s.$

So intuitively you have the sum of two arithmetic progressions of length $\delta-1,$ and $s+1,$ with a nontriviality property and this improves the bound.