We study a construction of the Mathieu group M12 using a game reminiscent of Loyd’s “15-puzzle”. The elements of M12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a “pseudogroup” M13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, a...

In the early 80-ties it became clear that the classification of the non-abelian finite simple groups was complete. Among the finite simple groups we find several families: the alternating groups and various families of groups of Lietype and their twisted analogues. Besides these families there exist 26 sporadic finite simple groups. In these notes we consider five sporadic simple groups, the Ma...

In this paper from q-ary perfect codes new completely regular q-ary codes are constructed. In particular, two new ternary completely regular codes are obtained from ternary Golay [11, 6, 5] code. The first [11, 5, 6] code with covering radius ρ = 4 coincides with the dual Golay code and its intersection array is (22, 20, 18, 2, 1; 1, 2, 9, 20, 22) . The second [10, 5, 5] code, with covering rad...

3-phase Golay sequence and array triads are a natural generalisation of 2-phase pairs, yet their study has until now been largely neglected. We present exhaustive search results for all lengths up to 24, showing that the existence pattern is much richer than pairs. give an elementary proof there no triad whose length congruent 4 modulo 6. two construction methods triads, show how project lower-...

In the paper the study of synthetic transmit aperture method applying the Golay coded transmission for medical ultrasound imaging is presented. Longer coded excitation allows to increase the total energy of the transmitted signal without increasing the peak pressure. Moreover signal-to-noise ratio and penetration depth are improved while maintaining high ultrasound image resolution. In the work...

Several existing strategies for estimating the axial intensity derivative in the transport-of-intensity equation (TIE) from multiple intensity measurements have been unified by the Savitzky-Golay differentiation filter--an equivalent convolution solution for differentiation estimation by least-squares polynomial fitting. The different viewpoint from the digital filter in signal processing not o...

For the Golay code GGolay with codelength n = 24 the star trellis was proposed in [1]. The time axis consists of a number of parts I = ∪jIj , which are joint in one common point. In Figure 1 an example factor graph representation is depicted. This example has three equal length parts. The time axis of the parts are given by Ij = {0, (j − 1)n/3 + 1, . . . , jn/3,∞}, where ∞ denotes the junction....

A pump signal based on bipolar pulse coding and single-sideband suppressed-carried (SSB-SC) modulation is proposed for Brillouin optical time-domain analysis (BOTDA) sensors. Making a sequential use of the Brillouin gain and loss spectra, the technique is experimentally validated using bipolar complementary-correlation Golay codes along a 100 km-long fiber and 2 m spatial resolution, fully reso...