Code Construction and Decoding Algorithms for Semi-Quantitative Group Testing with Nonuniform Thresholds

We analyze a new group testing scheme, termed semi-quantitative group testing, which may be viewed as a concatenation of an adder channel and a discrete quantizer. Our focus is on non-uniform quantizers with arbitrary thresholds. For the most general semi-quantitative group testing model, we define three new families of sequences capturing the constraints on the code design imposed by the choice of the thresholds. The sequences represent extensions and generalizations of Bh and certain types of super-increasing and lexicographically ordered sequences, and they lead to code structures amenable for efficient recursive decoding. We describe the decoding methods and provide an accompanying computational complexity and performance analysis

Explicit constructions of infinite families of MSTD sets

We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We conclude by generalizing our method to compare linear forms epsilon_1 A + ... + epsilon_n A with epsilon_i in {-1,1}.Comment: Version 2: 14 pages, 1 figure. Includes extensions to ternary forms and a conjecture for general combinations of the form Sum_i epsilon_i A with epsilon_i in {-1,1} (would be a theorem if we could find a set to start the induction in general

On the Structure of Sets of Large Doubling

We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $\Lambda(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.Comment: 23 pages, changed title, revised version reflects work of Meyer that we were previously unaware o

Constructions of Generalized Sidon Sets

We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa. We also further optimize Koulantzakis' idea of interleaving several copies of a Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia, and Habsieger & Plagne. The resulting constructions yield the largest known generalized Sidon sets in virtually all cases.Comment: 15 pages, 1 figure (revision fixes typos, adds a few details, and adjusts notation