Extremal Collections of $k$-Uniform Vectors
Abstract
We show any matrix of rank $r$ over $\mathbb{F}_q$ can have $\leq \binom{r}{k}(q-1)^k$ distinct columns of weight $k$ if $ k \leq O_q(\sqrt{\log r})$ (up to divisibility issues), and $\leq \binom{r}{k}(q-1)^{r-k}$ distinct columns of co-weight $k$ if $k \leq O_q(r^{2/3})$. This shows the natural examples consisting of only $r$ rows are optimal for both, and the proofs will recover some form of uniqueness of these examples in all cases.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2018
- DOI:
- 10.48550/arXiv.1801.09609
- arXiv:
- arXiv:1801.09609
- Bibcode:
- 2018arXiv180109609B
- Keywords:
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- Mathematics - Combinatorics;
- 05B35;
- 05C35
- E-Print:
- 14 pages, major corrections to Theorem 1.6 and Corollary 3.7