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n-ary Gray Code

The binary-reflected Gray code described above is invariably referred to as the ‘Gray code’. However, over the years, mathematicians have discovered other types of Gray code. One such code is the n-ary Gray code, also called the non-Boolean Gray code owing to the use of non-Boolean symbols for encoding. The generalized representation of the code is the (n, k-Gray code, where n is the number of independent digits used and k is the word length. A ternary Gray code (n = 3) uses the values 0, 1 and 2, and the sequence of numbers in the two-digit word length would be (00, 01, 02, 12, 11, 10, 20, 21, 22). In the quaternary (n = 4) code, using 0, 1, 2 and 3 as independent digits and a two-digit word length, the sequence of numbers would be (00, 01, 02, 03, 13, 12, 11, 10, 20, 21, 22, 23, 33, 32, 31, 30). It is important to note here that an (n, k-Gray code with an odd n does not exhibit the cyclic property of the binary Gray code, while in case of an even n it does have the cyclic property. The (n, k-Gray code may be constructed recursively, like the binary-reflected Gray code, or may be constructed iteratively. The process of generating larger word-length ternary Gray codes. The columns between those representing the ternary Gray codes give the intermediate steps.

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