Part X of a deep-dive series on LFSRs covers two practical applications: LFSR-based counters and absolute position encoders. For counters, the key challenge is decoding the count via discrete logarithms, which is easier when the LFSR period is smooth (composed of small prime factors). The article identifies degree-48 and degree-60 primitive polynomials as particularly well-suited, and explains how reducible polynomials (products of primitive polynomials with coprime periods) can construct near-maximal LFSRs with even smoother periods — up to 91 bits. For encoders, it introduces chain-code (De Bruijn sequence) based absolute encoders, showing how a single extra track (Y) alongside standard AB incremental encoder tracks can provide absolute position resynchronization after only a few counts, using LFSR state recovery and discrete logarithms — with no hardware cost increase over a traditional ABZ encoder.
