The Equivalence of Certain Harper Codes

In order to send numerical data over a binary channel, each input number must be encoded into a suitable binary sequence for transmission. For example, when a sampler and quantizer are used, a binary sequence is assigned to each quantization level. For each sample, the number of the appropriate quantization level is transmitted by sending the binary sequence assigned to the level. But how should the binary sequences be assigned? One approach is to use the natural binary representation of each number. Alternatively, a Gray code might be used with the idea that its unit-distance properties are in some sense desirable. If the transmission system is error-free and if the binary sequences are unique, it does not matter how the sequences are assigned. However, if transmission errors can occur, some assignment algorithms may be preferable to others. In this paper, the performance of certain binary encoding algorithms is considered. The average magnitude by 3113 3114 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1969