Freeman Slaughter | Searcharxiv

arXiv2026

The covering radius of RM(1,9) is precisely 242

Let $\text{RM}(r,n)$ be the $r$th order Reed-Mullet code of length $2^n$, and let $ρ(r,n)$ denote its covering radius. It is known that $242\leρ(1,9)\le 244$, and the determination of $ρ(1,9)$ is a long-standing open question. We give a computer-assisted proof that $ρ(1,9)=242$, leveraging the classification of the $349$ orbits in $\text{RM}(3,9)/\text{RM}(2,9)$ under $\text{GL}(9,2)$ [1,3]. The notebook to verify the computer search portion, Verify_RM(1,9).ipynb, contains all data and recomputes each check, and may be found in the repository https://github.com/FreemanSlaughter/Covering-Radius.

cs.IT↗

arXiv2026

A Bounded-Defect Construction for the Mondrian Art Problem

We construct an infinite family of solutions to the Mondrian art problem with absolute defect bounded above by $280$, of side lengths $n_0 = 315$, $n_1 = 106785$, and $n_{j+2} = 340n_{j+1} - n_j$, where $n_j$ is congruent to $315 \pmod{630}$. This absolute bound shows that the minimum defect does not necessarily increase with $n$, providing a counterexample to a conjecture which has been open since the problem's original proposal. Our design is derived from a well-known seven-rectangle dissection over the real quadratic field $\mathbb{Q}(\sqrt{19})$, with non-integer side lengths; we tailor the side lengths to be integers via an infinite Pell family. Using the same method, we devise a second family with defect $560$ that contains infinitely many side lengths which are divisible by any given positive integer, implying that for any $k$ there are infinitely many side lengths $n \equiv 0 \pmod{k}$ with defect bounded above by $560$. This constant bound offers a dramatic improvement on the previous quadratic bounds.

math.NT↗