We here study Max Hamming XSAT, ie, the problem of finding two XSAT models at maximum Hamming distance. By using a recent XSAT solver as an auxiliary function, an O(1.911^n) time algorithm can be constructed, where n is the number of variables. This upper time bound can be further improved to O(1.8348^n) by introducing a new kind of branching, more directly suited for finding models at maximum ...

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [26, 24]. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these...

We consider the problem of computing the Hamming weight of an n-bit vector using a circuit with gates for addition and multiplication modulo 2 (alternatively, XOR and conjunction gates) only. The number of multiplications necessary and sufficient to build such a circuit is called the “multiplicative complexity” of the Hamming weight function, and is denoted by c∧(H). We prove c∧(H) = n−HN(n) wh...

We estimate discrete Fourier transform, ambiguity, and Hamming-autocorrelation of m-ary sequences in terms of their (periodic) correlation measure of order 4. Roughly speaking, we show that every pseudorandom sequence, that is, any sequence with small correlation measure up to a sufficiently large order, cannot have a large discrete Fourier transform, ambiguity, or Hamming-autocorrelation. Conv...

When approximating binary similarity using the hamming distance between short binary hashes, we show that even if the similarity is symmetric, we can have shorter and more accurate hashes by using two distinct code maps. I.e. by approximating the similarity between x and x′ as the hamming distance between f(x) and g(x′), for two distinct binary codes f, g, rather than as the hamming distance be...

Tree-like partial cubes were introduced in [B. Brešar, W. Imrich, S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory, 23 (2003), 227–240] as a generalization of median graphs. We present some incorrectnesses from that article. In particular we point to a gap in the proof of the theorem about the dismantlability of the cube graph of a tree-like partial cube and ...

Given a language L, the Hamming distance of a string x to L is the minimum Hamming distance of x to any string in L. The edit distance of a string to a given language is deened similarly. First, we prove that the Hamming distance to languages in AC 0 is hard to approximate; it cannot be approximated within a factor O ? n 1 3 ? , for any > 0, unless P = NP (n denotes the length of the input stri...