5 problems

Let $P=\{a,b\}$ be a prefix-free binary block set, with block lengths $\ell_a=|a|\leqslant\ell_b=|b|$, and suppose that $K^{\star}(a,b)<\infty$. Two-block depth law. If…

For each integer $k\geqslant3$, define binary blocks … with lengths $\ell_a=|a_k|=2k$ and $\ell_b=|b_k|=2k+2$, and let $P_k=\{a_k,b_k\}$. The quantity $L_{P_k}$ denotes the relevan…

Let $q$ be the alphabet size and let $t$ be an arbitrary fixed number of transpositions. Consider codes of block length $n$ correcting $t$ transpositions, and let the optimal code…

Let $\eta\in(0,1)$ be an erasure probability, and let $\mathsf{BEC}(\eta)$ denote the binary erasure channel with erasure probability $\eta$. Let $C_{\mathsf{perm}}$ denote noisy p…

Let $P_{Z|X}$ be a strictly positive discrete memoryless channel (DMC), let $\hat{P}_{X_1^n}$ and $\hat{P}_{Y_1^n}$ denote the empirical distributions of the input and output seque…