Fractal digital sums and codes (Q1352418)

The aim of this paper is to study digital sums of the type \[ S_p(N)= \sum_{n<N}(- 1)^{\nu(pn)}, \] where \(\nu(k)\) denotes the sum of the binary digits of \(k\) and \(p\) is a given odd number. After a recent account of the subject, the case \(p=17\) is studied in detail: the function \(S_{17}\) is shown to be always positive and its fractal nature is discussed. Then the authors use these sum-of-digits functions to generate a special code: the analysis of this code shows that it is exponential, extremely redundant and thus strongly error correcting with a linear decoding procedure.