16 problems

Incorrect-path splitting conjecture. Under the Gaussian approximation, for any incorrect path that survives at some unfrozen bit $u_i$, it will split at some stage within…

Let $\mathcal{P}_{d}({\mathbb{F}}^n)$ be the class of polynomials of degree at most $d$ from ${\mathbb{F}}^n$ to ${\mathbb{F}}$. For a function $g:{\mathbb{F}}^n\to{\mathbb{F}}$, l…

Shangguan's full-rank conjecture. The matrix $M_{G,(J_1,\ldots,J_t)}$ has full column rank.

Let $L$ be a positive integer, let $C\geq1$ be a positive real, and let $R=\frac{k}{n}$ be the rate of an $[n,k]$ Reed–Solomon code. A code is list-decodable from radius $\rho$ wit…

For a positive integer $n$, write $[n]=\{1,\ldots,n\}$. For subsets $I_1,\ldots,I_t\subseteq[n]$, define their weight by … and write…

For a code of rate $R$ over a finite field, list-decodable from relative radius $r$ with list size $L$ means that every Hamming ball of radius $r n$ contains at most $L$ codewords.…

List decodability of Reed-Solomon codes up to capacity. For every $\rho>0$, there is a constant $C_\rho$ such that every Reed-Solomon code of length $n$ and rate $\rho$ is list-dec…

Let $\mathcal B$ be a symmetric set in $\bR^n$ of volume $V=2^{\mathcal O(n)}$, let $k=cn$ for a constant $0<c<1$, let $\mu$ be the Haar distribution on lattices, and let $Y$ be Po…

Let $q$ be a prime power, let $0<p<1-1/q$, and let $H_q(p)$ denote the $q$-ary entropy. A random linear code of rate $1-H_q(p)-\delta$ is chosen uniformly among linear codes of tha…

Consider a minimal convolutional code with $m$ memory elements and a degree-$v$ distance-spectrum-optimal CRC code, used together under serial list Viterbi decoding at a fixed SNR.…

Universal finite-simple-group conjecture. The class of finite simple groups is universally CombEcon.

Consider polar codes decoded by list decoding with a fixed list size $L$, and let $k$ denote the code dimension. Dimension-dependent SC-performance conjecture. The SC-performance o…

Consider two polar codes $A$ and $B$ with the same code length $n$ and message length $k$. Let ${\rm FER_{SC}}(\cdot)$ and ${\rm FER_{ML}}(\cdot)$ denote their frame-error rates un…

Consider two polar codes $A$ and $B$ with the same code length $n$ and message length $k$. Let ${\rm FER_{SC}}(\cdot)$ and ${\rm FER_{ML}}(\cdot)$ denote their frame-error rates un…

Correct-path survival conjecture. As $P_e(u_i)$ approaches zero, with high probability the current path survives at $u_i$ without splitting and $u_i$ is correctly decoded. If the s…

Gopalan–Klivans–Zuckerman conjecture. For fixed $d$ and finite field $\mathbb{K}$,