The efficient coding hypothesis assumes that biological sensory systems use neural codes that are optimized to best possibly represent the stimuli that occur in their environment. When formulating such optimization problem of neural codes, two key components must be considered. The first is what types of constraints the neural codes must satisfy? The second is the objective function itself -- what is the goal of the neural codes? We seek to provide a systematic framework to address these types of problem. Previous work often assume one specific set of constraint and analytically or numerically solve the optimization problem. Here we want to put everything in a unified framework and show that these results can be understood from a much more generalized perspective. In particular, we provide analytical solutions for a variety of neural noise models and two types of constraint: a range constraint which specifies the max/min neural activity and a metabolic constraint which upper bounds the mean neural activity. In terms of objective functions, most common models rely on information theoretic measures, whereas alternative formulations propose incorporating downstream decoding performance. We systematically evaluate different optimality criteria based upon the $L_p$ reconstruction error of the maximum likelihood decoder. This parametric family of optimal criteria includes special cases such as the information maximization criterion and the mean squared loss minimization of decoding error. We analytically derive the optimal tuning curve of a single neuron in terms of the reconstruction error norm $p$ to encode natural stimuli with an arbitrary input distribution. Under our framework, we can try to answer questions such as what is the objective function the neural code is actually using? Under what constraints can the predicted results provide a better fit for the actual data? Using different combination of objective function and constraints, we tested our analytical predictions against previously measured characteristics of some early visual systems found in biology. We find solutions under the metabolic constraint and low values of $p$ provides a better fit for physiology data on early visual perception systems.