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Let $$p$$ be an odd  prime, $q=p^e$, $$e \geq 1$$, and $$\mathbb{F} = \mathbb{F}_q$$ denote the finite field of $$q$$ elements.  Let $$f: \mathbb{F}^2\to \mathbb{F}$$ and  $$g: \mathbb{F}^3\to \mathbb{F}$$  be functions, and  let $$P$$ and $$L$$ be two copies of the 3-dimensional vector space $$\mathbb{F}^3$$. Consider a bipartite graph $$\Gamma_\mathbb{F} (f, g)$$ with vertex partitions $$P$$ and $$L$$ and with edges defined as follows: for every $$(p)=(p_1,p_2,p_3)\in P$$ and every $$[l]= [l_1,l_2,l_3]\in L$$, $$\{(p), [l]\} = (p)[l]$$ is an edge in $$\Gamma_\mathbb{F} (f, g)$$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question  appeared in Nassau: Given $$\Gamma_\mathbb{F} (f, g)$$,  is it always possible to find a function $$h:\mathbb{F}^2\to \mathbb{F}$$ such that the graph $$\Gamma_\mathbb{F} (f, h)$$  with the same vertex set as $$\Gamma_\mathbb{F} (f, g)$$ and with edges $(p)[l]$  defined in a similar way  by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $$\Gamma_\mathbb{F} (f, g)$$ for infinitely many $$q$$?  In this paper we show that the  answer to the question is negative and the graphs $$\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$$ provide such an example for $$p \equiv 1 \pmod{3}$$. Our argument is based on proving that the automorphism group of these graphs has order $$p$$, which is the smallest possible order of the automorphism group of graphs of the form $$\Gamma_{\mathbb{F}}(f, g)$$. 

The linear representation of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes. We consider the LDPC codes defined by taking the incidence matrix and its transpose as parity-check matrices, and in the former case prove a conjecture of Vandendriessche that the code is generated by words of minimum weight called plane words. In the latter case we compute the minimum weight in several cases and provide explicit constructions of minimum weight codewords.