33,165 research outputs found

    Distance-regular graphs

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    This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

    On unbalanced Boolean functions with best correlation immunity

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    It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in nn variables cannot exceed 2n/3βˆ’12n/3-1; moreover, it is 2n/3βˆ’12n/3-1 if and only if the function corresponds to an equitable 22-partition of the nn-cube with an eigenvalue βˆ’n/3-n/3 of the quotient matrix. The known series of such functions have proportion 1:31:3, 3:53:5, or 7:97:9 of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio C:BC:B of the number of ones and zeros, then CBCB is divisible by 33. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. We also establish that there are exactly 22 equivalence classes of the equitable partitions of the 1212-cube with quotient matrix [[3,9],[7,5]][[3,9],[7,5]] and 1616 classes, with [[0,12],[4,8]][[0,12],[4,8]]. These parameters correspond to the Boolean functions in 1212 variables with correlation immunity 77 and proportion 7:97:9 and 1:31:3, respectively (the case 3:53:5 remains unsolved). This also implies the characterization of the orthogonal arrays OA(1024,12,2,7)(1024,12,2,7) and OA(512,11,2,6)(512,11,2,6).Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse
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