On decimal sequences (Corresp.)

The properties of decimal sequences of rational numbers are investigated with the idea of using them as random sequences and error-correcting codes, and in other applications in communications. Several structural properties of decimal sequences are presented with special attention being given to binary decimal sequences and reciprocals of primes. For a certain class of decimal sequences of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l/q, q</tex> prime, it is shown that the digits spaced half a period apart add up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r - 1</tex> , where r is the base in which the sequence is expressed. Also for the same class all subsequences of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r^{m} &gt; q</tex> , are distinct. These underlying structural properties have made it possible to establish a lower bound on the Hamming distance between a given sequence and its cyclic shifts and also to obtain an upper bound on the autocorrelation function. A condition for the cross correlation of two decimal sequences being zero has also been obtained. The outcome of the calculations on decimal sequences of several primes is presented, and the results indicate that the autocorrelation function approximates that obtained for sequences of independent, equally likely random digits.