Geometric representations of linear codes

We say that a linear code CC over a field FF is triangular representable if there exists a two dimensional simplicial complex Δ such that CC is a punctured code of the kernel ker⁡Δker⁡Δ of the incidence matrix of Δ over FF and there is a linear mapping between CC and ker⁡Δker⁡Δ which is a bijection and maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p)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 CC. 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)GF(p), p 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.