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For aperiodic (sometimes also called, full period) strings, the term you are looking for is Lyndon words. These are the (unique lexicographically-least) representative of a full-period necklace (as st …

The concept you are referring to is a covering code. These have been extensively studied and a great reference would be the book "Covering Codes" by G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein. ( …

I can answer your first question fully, and the second question only partially. Question 1: Assuming you meant $\log_2$ in your expression, the answer is $h=1$. That is because $w(n)=\frac{1}{n}\sum_ …

Hi everyone, In the process of studying a problem in the Johnson association scheme I came across the following sum: $$\sum_{k\geq 0}(-1)^k\binom{n}{k}\binom{a-k}{a-b}\binom{c+k}{b}.$$ All the variab …

You can find a lot of information in the recent book: "Perfect Codes and Related Structures," by T. Etzion. The notes to Chapter 5 contain a long list of references. In particular, if you're intereste …

I'm trying to find out whether the following graph has a name: Let $W$ be an $n$-dimensional vector space over $GF(q)$. The vertices of the graph are all the subspaces of $W$. Two subspaces $W_1$ and …

What you are looking for is an $(r,s,n)$-covering design. A good starting point might be the La Jolla Covering Repository or in fact, any book on design theory. In general, the smallest possible size …

While considering eigenvalues of a certain Cayley graph, I came across the following sum: $$\sum_{r=0}^{d}\sum_{i=0}^{r} (-1)^{i} \binom{w}{i}\binom{n-w}{r-i}$$ where $d$, $w$, and $n$, are all positi …