18 problems

Let $m$ be a positive integer, let $h$ be as in Theorem 8, and let $\mathcal{C}_{(2,2^m+1,3,h)}$ denote the BCH code in that theorem. Its minimum distance is denoted by $d$. Minimu…

Ding's conjecture. The minimum distance of the primitive BCH code ${\mathbf{C}}_{(q,q^m-1,q^t+1,1)}$ always equals its Bose distance $d_B$.

Let $m$ and $t$ be integers, and let $C$ be obtained by adding an overall parity check to the primitive BCH code of length $2^m-1$ and designed distance $2t-1$. Thus $C$ has length…

Guruswami–Sudan conjecture for binary BCH codes. The Guruswami–Sudan algorithm should be modifiable so as to achieve the Johnson bound for binary BCH codes.

Let $m$ be a positive integer. For each integer $\delta$ with $2\leq \delta\leq 2^m-1$, let $\mathcal{C}_{(2,m,\delta)}$ be the binary primitive narrow-sense BCH code of length…

Ding et al.'s conjecture. The minimum distance is always equal to the Bose distance:

Let $\mathbf{H}$ be the parity-check matrix of a BCH code, and compare its row density with that of parity-check matrices used for LDPC codes. Density conjecture. The root cause of…

Let $d$ be the minimum distance and $d_B$ the BCH bound of a narrow-sense primitive BCH code. Upper-bound conjecture. The minimum distance satisfies … This conjecture concerns the…

Geng et al.'s AMDS conjecture. The code $\mathcal{C}^\perp_{(q,q+1,3,4)}$ is an AMDS code with parameters $[q+1,4,q-3]$.

Let $q=3^m$ with $m\geq 3$ odd, and let $\mathcal{C}_{(q,q+1,3,4)}$ denote the BCH code of length $q+1$ and designed distance $4$ over $\mathbb{F}_q$. A linear code of length $n$ a…

Let $s$ be an integer with $s\ge 6$ and $s$ even, and consider the BCH code $\mathcal{C}_{(2^s,2^s+1,3,4)}$. Binary BCH AMDS conjecture. The code $\mathcal{C}_{(2^s,2^s+1,3,4)}$ is…

Let $q=p^s$, where $p$ is an odd prime and $s\ge 2$, and set $h=\frac{p-1}{2}$. Let $\mathcal{C}_{(q,q+1,3,h)}$ denote the BCH code in question. BCH-code conjecture. The code…

Let $q$ be an odd prime power, let $m$ be an even positive integer, and set $n=q^m+1$. Let $u2(m)$ denote the 2-adic valuation of $m$, let $deltai$ be the $i$-th largest coset lea…

Let $q$ be a power of $2$ and let $n=q^m+1$, with the quantities $\delta_2$ and $\delta_3$ defined as in the preceding cases for even $m$. Second and third largest coset leader con…

Let $q$ be an odd prime power and let $n=q^m+1$, with the quantities $\delta_3$ and $\delta_4$ defined as in the preceding cases for even $m$. Third and fourth largest coset leader…

Let $q$ be odd and $m \geq 2$. For an integer $t$ with $1 \leq t \leq \bar m$, let $\delta=(q^t-1)/2$ and let ${\mathcal{C}}_{(q,n,2\delta,\frac{n}{2}-\delta+1)}$ be the code from…

Conjecture on the two minimum distances. The two codes have minimum distances

Uniqueness conjecture. When $2k=q$, the only recursive MDS matrices that exist come from these shortened BCH codes.