On the normality of pp-ary bent functions

Depending on the parity of $n$ and the regularity of a bent function $f$ from $\mathbb F_p^n$ to $\mathbb F_p$, $f$ can be affine on a subspace of dimension at most $n/2$, $(n-1)/2$ or $n/2- 1$. We point out that many $p$-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) $n/2$-normal, i.e. affine on a $n/2$-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not $n/2$- normal. We develop an algorithm for testing normality for functions from $\mathbb F_p^n$ to $\mathbb F_p$. Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.Comment: 13 page

Wave Front Sensing and Correction Using Spatial Modulation and Digitally Enhanced Heterodyne Interferometry

This thesis is about light. Specifically it explores a new way sensing the spatial distribution of amplitude and phase across the wavefront of a propagating laser. It uses spatial light modulators to tag spatially distinct regions of the beam, a single diode to collect the resulting light and digitally enhanced heterodyne interferometry to decode the phase and amplitude information across the wavefront. It also demonstrates how using these methods can be used to maximise the transmission of light through a cavity and shows how minor aberrations in the beam can be corrected in real time. Finally it demonstrate the preferential transmission of higher order modes. Wavefront sensing is becoming increasingly important as the demands on modern interferometers increase. Land based systems such as the Laser Interferometer Gravitational-Wave Observatory (LIGO) use it to maximise the amount of power in the arm cavities during operation and reduce noise, while space based missions such as the Laser Interferometer Space Antenna (LISA) will use it to align distant partner satellites and ensure that the maximum amount of signal is exchanged. Conventionally wavefront sensing is accomplished using either Hartmann Sensors or multi-element diodes. These are well proven and very effective techniques but bring with them a number of well understood limitations. Critically, while they can map a wavefront in detail, they are strictly sensors and can do nothing to correct it. Our new technique is based on a single-element photo-diode and the spatial modulation of the local oscillator beam. We encode orthogonal codes spatially onto this light and use these to separate the phases and amplitudes of different parts of the signal beam in post processing. This technique shifts complexity from the optical hardware into deterministic digital signal processing. Notably, the use of a single analogue channel (photo-diode, connections and analogue to digital converter) avoids some low-frequency error sources. The technique can also sense the wavefront phase at many points, limited only by the number of actuators on the spatial light modulator in contrast to the standard 4 points from a quadrant photo-diode. For ground-based systems, our technique could be used to identify and eliminate higher-order modes, while, for space-based systems, it provides a measure of wavefront tilt which is less susceptible to low frequency noise. In the future it may be possible to couple the technique with an artificial intelligence engine to automate more of the beam alignment process in arrangements involving multiple cavities, preferentially select (or reject) specific higher order modes and start to reduce the burgeoning requirements for human control of these complex instruments