Single-track Gray codes

摘要

A common use of Gray codes is in reducing quantisation errors invarious types of analogue-to-digital conversion systems. As a typicalexample, a length n Gray code can be used to record the absolute angularpositions of a rotating wheel by encoding the codewords on nconcentrically arranged tracks. A number of reading heads, n, mountedradially across the trades, suffice to recover the codewords and it iswell known that quantisation errors are minimised by using a Grayencoding. When a high resolution is required, the need for a largenumber of concentric tracks results in encoders with large physicaldimensions. This poses a problem in the design of small-scale orhigh-speed devices. We propose single-trade Gray codes as a way ofovercoming this problem. Let W₀,W₁...,W_p-1 be the codewords of a Gray code C and writeW_i=[w_i0,w_i1,...,w_in-1]T. We call the sequence w₀j,w₁j,...,w_p-1j component sequence j of C