Block codes, which correct asymmetric errors with limited-magnitude, are studied. These codes have been applied recently for error correction in flash memories. The codes will be represented by lattices and the constructions will be based on a generalization of Sidon sequences. In particular we will consider perfect codes for these type of errors.

Fix integers b > a ≥ 1 with g := gcd(a, b). A set S ⊆ N is {a, b}-multiplicative if ax 6= by for all x, y ∈ S. For all n, we determine an {a, b}-multiplicative set with maximum cardinality in [n], and conclude that the maximum density of an {a, b}-multiplicative set is b b+g . Erdős [2, 3, 4] defined a set S ⊆ N to be multiplicative Sidon1 if ab = cd implies {a, b} = {c, d} for all a, b, c, d ∈...

Let k > 1 be an integer. A set A ⊂ Z is a k-fold Sidon set if A has only trivial solutions to each equation of the form c1x1 + c2x2 + c3x3 + c4x4 = 0 where 0 6 |ci| 6 k, and c1 + c2 + c3 + c4 = 0. We prove that for any integer k > 1, a k-fold Sidon set A ⊂ [N ] has at most (N/k)1/2 + O((Nk)1/4) elements. Indeed we prove that given any k positive integers c1 < · · · < ck, any set A ⊂ [N ] that c...

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solvability of some equations and the distribution of their solutions. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.

Here is a direct problem for Sidon sets: Given linear form $$\varphi = c_1 x_1 + \cdots c_h x_h $$ , construct and describe sets A that are . This paper considers an inverse set A, determine the forms such