Let ∆ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex ∆s of ∆ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, . . . , xn] of the symmetric algebraic shifted, exterior algebraic shifted and combinatorial shifted complexes of ∆ are equal.

We study binary linear codes C obtained from the quadric Veronese embedding of P3 in P9 over F2. show how one can find higher weight spectra these codes. Our method will be a Stanley-Reisner rings series matroids associated to each code C.

Gu map-1 is a modified version of GGH map. It uses same ideal lattices for constructing the trapdoors, while the novelty is that no encodings of zero are given. In this short paper we show that Gu map-1 cannot be used for the instance of witness encryption (WE) based on the hardness of 3-exact cover problem. That is, if Gu map-1 is used for such instance, we can break it by soving a combined 3-...