In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length n with complementary autocorrelation is at worst a 2 3n 2 −6 problem. Andres, in his 1977 master’s thesis, developed an algorithm which reduced this to a 2 n 2 −1 search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a 2 n 2 ...

pollution has become a very serious threat to our environment. monitoring pollution is the rst step toward planning to save the environment. the use of dierential equations, monitoring pollution has become possible. in this paper, a ritz-collocation method is introduced to solve non-linear oscillatory systems such as modelling the pollution of a system of lakes. the method is based upon bernoul...

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V1, V2 in two different variables, these kernels may be expressed in terms of finite dimensional “windows” s...

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall-Littlewood polynomials (spher...

A new proof of the uniqueness and of the existence of the extended ternary Golay code is presented. The proof connects the code to the projective plane of order 3 and is of an elementary nature. The available proofs of the uniqueness of the extended ternary Golay code [2,7] are much more complicated than the standard corresponding proof in the binary case [2]. The prevailing opinion seems to be...

When can one see from the spectrum of a graph whether it is distance-regular or not? We give some new results for when this is the case. As a consequence we find (among others) that the following distance-regular graphs are uniquely determined by their spectrum: The collinearity graphs of the generalized octagons of order (2, 1), (3, 1) and (4, 1), the Biggs-Smith graph, the M22 graph, and the ...

The extended Golay code is shown to be representable as a chained polar subcode. This enables its decoding with the successive cancellation decoding algorithm and its stack generalization. The decoder can be further simplified by employing fast Hadamard transform. The complexity of the obtained algorithm is comparable with that of the Vardy algorithm.

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W , such that S (but not necessarily R) is reduced. For each such pair (R, S) we construct a family of W -invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters q, t1, t2, . . . , tr, where r (= 1, 2 or 3) is the number of W -orbits in...

AbstTactwe present algorithms for maximum likelihood soft decoding of the second order Reed-Muller codes RM(2,m) and the extended (24,12,8) Golay code. The decoding procedures are based on a concise representation of appropriately selected cosets of a subcode of the code considered. Such representation enables application of certain elimination rules. A structure, called constrained design, is ...

In [17] Turyn constructed the famous binary Golay code of length 24 from the extended Hamming code of length 8 (see also [10, Theorem 18.7.12]). The present note interprets this construction as a sum of tensor products of codes and uses it to construct certain new extremal (or at least very good) self-dual codes (for example an extremal doubly-even binary code of length 80). The lattice counter...