Let $\mathcal{F}$ be a k‐uniform set system defined on a ground set of size n with no singleton intersection; i.e., no pair $A,B\in\mathcal{F}$ has $|A\cap B|=1$. Frankl showed that $|\mathcal{F}|\leq\binom{n-2}{k-2}$ for $k\geq4$ and n sufficiently large, confirming a conjecture of Erdo˝s and Sós. We determine the maximum size of $\mathcal{F}$ for $k=4$ and all n, and also establish a stability result for general k, showing that any $\mathcal{F}$ with size asymptotic to that of the best construction must be structurally similar to it.

The covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword. The authors introduce a new function l(m,r), called the length function, which equals the smallest length of a binary code of codimension m and covering radius r. They investigate basic properties of the length function. Projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l(m,2) obtained by amalgamation of Hamming codes. General methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l(m,2). A table is presented which gives the best results now known for l(m,r) with m>

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length< 63(with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of length\geq 63.

We show that binary cyclic codes of lengthn = 2^{m} - 1with generatorg(x)=m_{1}(x)m_{7}(x)have minimum distance< 5unlessm = 5(and possiblym = 11, 13, or 17).

Simple descriptions of Preparata and Goethals codes are provided.