The Golden mean, Fibonacci matrices and partial weakly super-increasing sources

A source S ¼ fs1; s2; . . .g, with at least i þ 1 source symbols, having a binary Huffman code with codeword lengths satisfying l1 ¼ 1; l2 ¼ 2; . . . ; li ¼ i, is called an i-level partial weakly super-increasing (PWSI) source. Connections between these sources, Fibonacci matrices and the Golden mean are studied. It is shown that the Euclidean projection of the distributions associated with these sources is given by Fibonacci–Hessenberg matrices. While there is no upper bound on the expected codeword length of Huffman codes representing PWSI sources (and hence no upper bound on their entropy), the Fibonacci sequence and the Golden mean 1þ ffiffi 5 p 2 provide a lower bound on the maximum expected codeword length of these codes.