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Center for Nonlinear Studies |
We say that a linear code C is triangular representable if there exists a two dimensional simplicial complex $\Delta$ such that C is a punctured code of the kernel of the incidence matrix of $\Delta$ and there is a bijection between C and $\ker \Delta$ which maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p), where p is a prime, are triangular representable. In the case of finite fields, we show that this representation determines the weight enumerator of C. We present one application of this result to the partition function of the Potts model. On the other hand, we show that there exist linear codes over any field different from rationals and GF(p), p is a prime, that are not triangular representable. We show that every construction of triangular representation fails on a very weak condition that a linear code and its triangular representation have to have the same dimension.
Host: Misha Chertkov, chertkov@lanl.gov, 665-8119