Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)

The previous are identical to the expressions (2) and (3). Thus the n 1 columns do consist of two repetitions of 2”-r-level Gray code. Now if we combine the first two columns again, we reduce each 2”-‘-level Gray code into two 2n-2-level Gray codes, or, the complete array into four 2”-2-level Gray codes. This can continue until we have only m columns, which would be 2n-m repetitions of 2”‘-level Gray code. We have thus derived the lemma.