Correction of errors in multilevel Gray coded data

In this paper we present a new error-control technique intended for use in
2^l
-level data-transmission systems that employ Gray coding to transform a binary source sequence into the
2^l
-ary transmitted sequence. The codes, which we call
i
-compressed codes, make use of the structure of binary codes and have the property that for some integer
i
,
1 \leq i \leq l
, transmission errors can be corrected if the erroneously received signals lie less than
2^{i-1}
levels from the corresponding correct, or nominal signal levels. The number of such errors that can be corrected is related to the error-correcting capability of the underlying binary code used in the construction. In return for this restriction on the magnitude of correctable errors in the received signal, these codes have higher rates than binary codes of comparable length (in bits) and number of correctable errors. Hence in applications where it can be assumed that the fraction of errors exceeding a certain magnitude is negligible (or at least tolerable), this technique is more efficient than the conventional practice of placing a binary encoder between the data source and modulator and a binary decoder between the demodulator and data sink. Furthermore, although the
i
-compressed codes are nonbinary, the decoding algorithm is that of the underlying binary code plus a small amount of additional processing; hence it is generally simpler to implement than other nonbinary decoding algorithms. It is also observed that the rate of an
i
-compressed code is always greater than that of the underlying binary code. Thus certain classes of low-rate binary codes that have simple decoding algorithms can be used as underlying codes in the construction of high-rate easily decodable
i
-compressed codes. Finally, for the case
i = 1
, encoding and decoding becomes exceptionally simple and for this case it is possible to make use of "soft decisions" at the receiver to improve the performance.