Single-track Gray codes

A common use of Gray codes is in reducing quantisation errors in various types of analogue-to-digital conversion systems. As a typical example, a length n Gray code can be used to record the absolute angular positions of a rotating wheel by encoding the codewords on n concentrically arranged tracks. A number of reading heads, n, mounted radially across the trades, suffice to recover the codewords and it is well known that quantisation errors are minimised by using a Gray encoding. When a high resolution is required, the need for a large number of concentric tracks results in encoders with large physical dimensions. This poses a problem in the design of small-scale or high-speed devices. We propose single-trade Gray codes as a way of overcoming this problem. Let W₀,W₁...,W_p-1 be the codewords of a Gray code C and write W_i=[w_i⁰,w_i¹,...,w _i^n-1]^T. We call the sequence w₀ ^j,w₁^j,...,w_p-1^j component sequence j of C