Search Results

One of the more convenient and popular approximations of the sum is $$\frac{2^{nH(\frac{k}{n})}}{\sqrt{8k(1-\frac{k}{n})}} \leq \sum_{i=0}^k\binom{n}{i} \leq 2^{nH(\frac{k}{n})}$$ for $0< k < \frac{ …

To give an idea of how real-life applications may arise outside mathematics, let's consider a "testing" problem of some sort. I'll set up a problem to solve first, so Sperner's theorem doesn't appear …

I think it is an optimal version of a covering with index $1$. A $t$-$(n,k,\lambda)$ covering is an ordered pair $(U,\mathcal{B})$ of a finite set $U$ of cardinality $n$ and a finite set $\mathcal{B} …

As mentioned in Hao Chen's answer, what you're looking for seems to be a good mixed code. There don't seem to be many papers on this. But apparently the following paper gives the best known general up …

Edit: The possible "gap" of sort in Caro and Yuster's proof of their upper bound has just been fixed! See Ben Barber's comment below (and his joint paper with Daniela Kühn, Allan Lo and Deryk Osthus o …

Lucia's answer given in the linked question from the comment gives an upper bound (i.e., no pair should appear twice as a subset of $A_i$). But of course, the real question starts from here: When can …

It is called perfect hash families in the design theory and computer science literature. A perfect hash family PHF$(N; k, v, t)$ is an $N \times k$ array on $v$ symbols with $v \geq t$, where for eve …

Unfortunately, no. It is known that the maximum number of mutually disjoint $S(4,5,11)$s on the same point set is $2$. Any such pair are always isomorphic. So, you can't find $7$ disjoint copies of an …

If you regard each row of your desired matrix as a sequence of bipolar signals and each column as a time frame, then, with one additional condition that the matrix is circulant, what you're asking bec …

I'm not sure exactly what you mean because if you prove that there exists a cylcic $(v, k, \lambda)$-difference set for all $v$ except those that are excluded by known nonexistence results, you actual …

For the generalization in the first direction, the $p$-rank of the incidence matrix $N$ of an $S(2,k,v)$ is lower bounded by the dimension of the Steinberg module: $$\operatorname{rank}_2(N)(\operato …

An elementary counting argument shows that $2$-$(v,3,3)$ exists only if $v$ is odd (or, more precisely, for $\lambda \equiv 3 \pmod{6}$ a $2$-$(v,3,\lambda)$ exists only if $v \equiv 1 \pmod{2}$). Thi …

I guess you exclude trivial MDS codes, generalized Reed-Solomon codes, and MDS codes that can be obtained by code extension. If you exclude them all, there are still a bunch of MDS codes. In general, …

I guess you already know this, but I just stumbled on this paper that studies the exact same game you described: M. Fukuyama, A Nim game played on graphs, Theoret. Comput. Sci. 304 (2003), 387–399. …

Before answering your question, not every Hamming code is equivalent to some cyclic code. For instance, the ternary $[4,2,3]_3$ Hamming code (aka the tetracode) is not equivalent to any cyclic code. …