Some complementary Gray codes

TL;DR

This paper introduces generalized complementary Gray codes for various combinatorial objects, including binary, q-ary tuples, combinations, and permutations, relaxing conditions to construct codes in broader cases.

Contribution

It extends the concept of complementary Gray codes to new settings, including odd-sized binary tuples and Lee metric q-ary tuples, and presents the first cyclic Gray code for certain multiset permutations.

Findings

Constructed Gray codes for binary n-tuples with odd n.

Developed Gray codes for Lee metric q-ary n-tuples with odd n and even q.

Presented the first cyclic Gray code for permutations of specific multisets.

Abstract

A complementary Gray code for binary n-tuples is one that, when all the tuples are complemented, is identical to itself; this is equivalent to the complement of the first half of the code being identical to the second half. We generalize the notion of complementary to q-ary n-tuples, fixed size combinations of an n-set and permutations and, in each case, construct complementary Gray codes. We relax, as weakly as possible, the notions of complementary to cases where necessary conditions for existence are violated and construct Gray codes within the weakened definitions: these include binary n-tuples when n is odd and Lee metric q-ary n-tuples when n is odd and q is even. Finally a lemma used in the construction for permutations offers the first known cyclic Gray code for the permutations of a particular family of multisets.