I was working with BCH codes for Compressed Sensing. Through the simulation of BCH codes, I found some interesting facts which are useful for my Ph.D. thesis. I don't have mathematical proof of my findings regarding the BCH codes. Therefore I am proposing two conjectures. I want to see if anyone of you can prove or disprove my conjecture regarding the BCH codes. Now I begin stating my conjectures.
Conjecture 1 : For every integer $m\geq 3$,
(1) if $m$ is even, then there exists a BCH code of block length $2^m -1$, with minimum distance $2^{m-1}- 2^{\frac{m}{2}- 1}$, and the weight of codewords will be $0,\ 2^{m-1}-2^{\frac{m}{2} - 1},\ 2^{m -1},\ 2^{m - 1} + 2^{\frac{m}{2} - 1}$.
(2) If $m$ is odd, then there exists a BCH code of block length $2^{m}-1$, with minimum distance $2^{m-1}-2^{\frac{m+1}{2} - 1}$, and the weight of the codewords will be $0,\ 2^{m-1}-2^{\frac{m+1}{2} -1},\ 2^{m-1},\ 2^{m-1} + 2^{\frac{m+1}{2} - 1}$.
Conjecture 2: For every integer $m\geq 4$,
(1) if $m$ is even, then there exists a BCH code of block length $2^{m}-1$, with minimum distance $2^{m-1} - 2^{\frac{m}{2}}$, and the weight of the codewords will be $0,\ 2^{m-1}- 2^{\frac{m}{2}},\ 2^{m-1}- 2^{\frac{m}{2}} + 2^{\frac{m}{2} - 1},\ 2^{m-1},\ 2^{m-1} + 2^{\frac{m}{2} - 1},\ 2^{m-1}+ 2^{\frac{m}{2}}$.
(2) If $m$ is odd, then there exists a BCH code of block length $2^{m}-1$, with minimum distance $2^{m-1} - 2^{\frac{m+1}{2}}$, and the weight of the codewords will be $0,\ 2^{m-1}- 2^{\frac{m+1}{2}},\ 2^{m-1}- 2^{\frac{m+1}{2}} + 2^{\frac{m+1}{2} - 1},\ 2^{m-1},\ 2^{m-1} + 2^{\frac{m+1}{2} - 1},\ 2^{m-1}+ 2^{\frac{m+1}{2}}$.
I am proposing these two conjectures based on my simulation results. In simulations, I have verified these two conjectures for $m = 3$ to $20$. These two conjectures are so important to my Ph.D. thesis, and I will get one more research paper based on these two conjectures. Please let me know if you know any mathematical proof of my conjectures. I would be very grateful to you.
Obligatory disclosure: I am pursuing a Ph.D. in Electrical Engineering. My thesis work is on Compressed Sensing. I am using the BCH codes to design a Measurement Matrix for compressed sensing.
See for example the paper Boztas and Kumar, Binary Sequences with Gold-like Correlation but Larger Linear Span, IEEE Transactions on Information Theory 40.2 (1994), 532-537, full-text available here.
The sequence family is simply the BCH code with only one codeword taken from each cyclic equivalence class.
Theorem 1 proves the case for $m$ odd. The authors did not bother proving $m$ even, since in that case the solution is not optimal with respect to the Welch bound.
By the way, one should just state case 1 for $m=3$ separately, the second conjecture is really the same as the first other than $m=3.$
An earlier related reference is the University of Illinois report by T. Kasami from 1966 available here.