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We prove that every nontrivial linear $[n,k]$ maximum distance separable (MDS) code over a finite field $\F_q$ of characteristic two satisfies $n\le q+1$, except in dimensions $k=3$ and $k=q-1$, where $n\le q+2$. This re...
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For every prime power $q>19683$, we settle the Cheng--Murray deep-hole problem for full-length Reed--Solomon codes in every dimension $2\le k\le q-2$, with the characteristic-two cases stated explicitly. For odd $q$, the...
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We prove the Golomb--Welch conjecture for lattice tilings: for every $n\ge3$ and $r\ge2$, the integer lattice $\Z^n$ admits no lattice tiling by Lee balls $B_n(r)$ of radius $r$. Together with the classical constructions...