Applied Algebra, Algebraic Algorithms and Error-Correcting by G. David Forney Jr. (auth.), Marc Fossorier, Hideki Imai,

By G. David Forney Jr. (auth.), Marc Fossorier, Hideki Imai, Shu Lin, Alain Poli (eds.)

This e-book constitutes the refereed complaints of the nineteenth overseas Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-13, held in Honolulu, Hawaii, united states in November 1999.
The forty two revised complete papers offered including six invited survey papers have been conscientiously reviewed and chosen from a complete of 86 submissions. The papers are geared up in sections on codes and iterative deciphering, mathematics, graphs and matrices, block codes, jewelry and fields, interpreting equipment, code building, algebraic curves, cryptography, codes and interpreting, convolutional codes, designs, deciphering of block codes, modulation and codes, Gröbner bases and AG codes, and polynomials.

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1) {i,j}∈G which define a complex FG of free K[x, y, z]-modules in analogy with Eq. 4). , mij = mR for any edge ij of a region R. The graph G will be far from unique for monomial ideals M which are not generic. Example. Consider the mth power of the maximal ideal x, y, z , that is, M = x, y, z m = xi y j z k : i, j, k ∈ N, i + j + k = m . 2) The staircase of M is depicted in Figure 3(a) for m = 5. The graph in Figure 3(b) represents a free resolution of M for m = 5. This graph is essentially the edge graph of the hull polytope PM of M, as defined in Section 3.

Theorem 4. For every 3-connected planar graph G there exists a monomial ideal M in 3 variables which is minimally resolved by the bounded regions of G. 1] which states that 3-connected planar graphs are the edge graphs of 3-dimensional convex polytopes. 1, pp. 128], as explained in [3, §6]. The general non-triangulated case is more difficult. 1]. The complete proof of Theorem 4 is “under construction” and will be published elsewhere. In Figure 4 is a non-trivial example illustrating the encoding of a planar graph G by a monomial ideal M.

Note that the order 8 (dihedral group) symmetry of the graph cannot be reproduced in the monomial ideal. The square is realized by having an irreducible component which is determined by four surrounding generators, two of which have one coordinate in common. Similarly, the hexagons have six generators spread around the corresponding irreducible component, with each such generator sharing one coordinate with the irreducible component. Only the artinian components—those with generators on all three sides—define regions in G.

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