Bent functions, SDP designs and their automorphism groups

PhD ThesisIn a 1976 paper Rothaus coined the term “bent” to describe a function f from a vector space V (n, 2) to F2 with the property that the Fourier coefficients of (−1)f have unit magnitude. Such a function has the maximum possible distance from the set of linear functions, hence the name, and has useful correlation properties. These lead to various applications to coding theory and cryptography, some of which are described. A standard notion of the equivalence of two bent functions is discussed and related to the coding theory setting. Two constructions mentioned by Rothaus and generalised by Maiorana are described. A further generalisation of one of these, involving sets of bent functions on direct summands of the original vector space, is described and proved. Various methods including computer searches are used to find appropriate sets of bent functions and hence many new equivalence classes of bent functions of 8 variables. Equivalence class invariants are used to show that most of these classes cannot be constructed by the earlier methods. Some bounds on numbers of bent functions are discussed. A 2-design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block — such a design is very close to being a 3-design. All SDP designs are induced by bent functions, and conversely. Work on the automorphism groups of various SDP designs involving computer algebra is described. An SDP design on 256 points with trivial automorphism group is noted. Some connections with strongly-regular graphs are discussed. An infinite class of pseudo-geometric strongly-regular graphs induced by bent functions is noted, and bent functions which are their own Fourier transform duals are investigated. Finally, some open problems and ideas for future work are described

Antigen and checkpoint receptor engagement recalibrates T cell receptor signal strength

How T cell receptor (TCR) signal strength modulates T cell function and to what extent this is modified by immune checkpoint blockade (ICB) are key questions in immunology. Using Nr4a3-Tocky mice, we characterized early quantitative and qualitative changes that occur in CD4(+) T cells in relation to TCR signaling strength. We captured how dose- and time-dependent programming of distinct co-inhibitory receptors rapidly recalibrates T cell activation thresholds and visualized the immediate effects of ICB on T cell re-activation. Our findings reveal that anti-PD1 immunotherapy leads to an increased TCR signal strength. We defined a strong TCR signal metric of five genes upregulated by anti-PD1 in T cells (TCR.strong), which was superior to a canonical T cell activation gene signature in stratifying melanoma patient outcomes to anti-PD1 therapy. Our study therefore reveals how analysis of TCR signal strength—and its manipulation—can provide powerful metrics for monitoring outcomes to immunotherapy