Optimal Quasi-Gray Codes: Does the Alphabet Matter?

A quasi-Gray code of dimension n and length l over an alphabet Σ is a sequence of distinct words w1, w2, . . . , wl from Σ n such that any two consecutive words differ in at most c coordinates, for some fixed constant c > 0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word wi into its successor wi+1. We present construction of quasi-Gray codes of dimension n and length 3 over the ternary alphabet {0, 1, 2} with worst-case read complexity O(log n) and write complexity 2. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasiGray codes of dimension n and length at least 2 − 20n with worst-case read complexity 6 + logn and write complexity 2. Our results significantly improve on previously known constructions and for the odd-size alphabets we break the Ω(n) worst-case barrier for space-optimal (non-redundant) quasiGray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. ’14, Ben-Or and Cleve ’92, Barrington ’89, Coppersmith and Grossman ’75]. We also establish certain limits of our technique in the binary case. Although our techniques cannot give space-optimal quasi-Gray codes with small read complexity over the binary alphabet, our results strongly indicate that such codes do exist. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 616787. The third author was also partially supported by the Center of Excellence CE-ITI under the grant P202/12/G061 of GA ČR. [email protected] [email protected] [email protected] [email protected]