Joseph Swernofsky | Searcharxiv

arXiv2026

Salami Slicing Trellis for Synchronization Errors in DNA Coding

On top of substitution errors, DNA storage channels suffer from both insertions and deletions at the same time. It is therefore important to develop error-correcting codes with efficient encoders and decoders that can combat all three types of noise. This paper introduces the salami-slicing trellis, a decision-feedback trellis that computes bitwise posterior probabilities along each strand and is coupled with polar codes across strands. The decoder alternates between advancing the trellises by one position and polar-decoding the resulting cross-strand slice, feeding the decoded bits back to the trellises for the next position. Simulations suggest that the resulting coding scheme approaches the conjectured capacity of the substitution-insertion-deletion channel.

cs.IT↗

arXiv2019

$d$-Galvin families

The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S \cap A| = |S \cap \overline{A}|$. We consider a generalization of this question that comes from a possible approach in complexity theory. In the generalization the required property is, for any $A$, to be able to find $d$ sets from a family $\mathcal{F} \subseteq \binom{[n]}{n/d}$ that form a partition of $[n]$ and such that each part is balanced on $A$. We construct such families of size polynomial in the parameters $n$ and $d$.

math.CO↗