Toward a theory of avalanche-creating and error-controlling encoding schemes

This paper introduces a theory of avalanche-creating and error-controlling encoding schemes. We consider a Hamming space {0, 1} ⁿ and the problem of finding a bijection g : {0, 1}ⁿ rarr {0, 1}ⁿ, with inverse h = g^-1, that creates avalanche and controls error. That is, for any vectors u and v that are at most b₁ apart, their images g(u) and g(v) are at least b₂ apart; and for any vectors v and e, the distance between h(g(v) + e) and v is at most the length of e (the distance from e to 0ⁿ) plus b₃. It appears that the two objectives of achieving avalanche effect and error control simultaneously in a single encoding scheme are contradictory. This paper is the first attempt at understanding the tradeoffs. Toward this end, the paper determines nontrivial cases when desired encoding schemes can be efficiently constructed, cases when schemes exist but it is not known how to construct them efficiently, and cases when schemes do not exist at all