Quasi-realistic heterotic-string models in the free fermionic formulation typically contain an anomalous U (1), which gives rise to a Fayet-Iliopolous term that breaks supersymmetry at the one-loop level in string perturbation theory. Supersymmetry is restored by imposing F-and D-flatness on the vacuum. In [14] we presented a three generation free fermionic standard-like model which did not admit stringent F-and D-flat directions, and argued that the all the moduli in the model are fixed. The particular property of the model was the reduction of the untwisted Higgs spectrum by a combination of symmetric and asymmetric boundary conditions with respect to the internal fermions associated with the compactified dimensions. In this paper we extend the analysis of free fermionic models with reduced Higgs spectrum to the cases in which the SO(10) symmetry is left unbroken, or is reduced to the flipped SU (5) subgroup. We show that all the models that we study in this paper do admit stringent flat directions. The only examples of models that do not admit stringent flat directions remain the strandard-like models of ref. [14].

Using software under development at Baylor University, we explicitly construct all layer 1 gauge, weakly coupled free fermionic heterotic string models up to order 22 in four large space-time dimensions. The gauge models consist primarily of gauge content making a systematic construction process efficient. We present an overview of the model building procedure, redundancies in the process, methods used to reduce such redundancies and statistics regarding the occurrence of various combinations of gauge group factors and GUT groups. Statistics for both N = 4 and N = 0 models are presented.

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Using Baylor University's C++ software for construction of weakly coupled free fermionic heterotic string models, called the FF Framework, we explicitly construct the level 1 Kač-Moody ten dimensional heterotic string models with the aim of understanding the redundancies endemic to this construction method. We show that for models in any even number of large space-time dimensions with a massless left mover and an odd ordered right mover, the maximal number of space-time supersymmetries are present. We show that in order to produce all of the models for a given order, different basis vectors must be built; one cannot vary only the GSO coefficients. We also show that all combinations of two order-2 basis vectors do not produce the same models as all possible single order-4 basis vectors, implying the product of the orders used in a search does not necessarily determine the models built. We also show that to build all of the D=10 level-1 models the inputs must be: sets of single order-6 basis vectors, pairs of basis vectors with orders 3 and 2, or sets of three order-2 basis vectors.

The discovery that the number of physically consistent string vacua is on the order of 10 500 has prompted several statistical studies of string phenomenology. Contained here is one such study that focuses on the Weakly Coupled Free Fermionic Heterotic String (WCFFHS) formalism. Presented are systematic extensions of the well-known NAHE (Nanopoulos, Antoniadis, Hagelin, Ellis) set of basis vectors, which have been shown to produce phenomenologically realistic models. Statistics related to the number of U (1)'s, the specific gauge groups and their factors, non-Abelian singlets, and spacetime supersymmetries (ST SUSYs) are discussed for the full range of models produced.These findings are compared with prior results of other large-scale investigations. Statistical coupling between the gauge groups and the number of ST SUSYs is also discussed. In particular, for order-3 extensions there is a correlation between the appearance of exceptional groups and enhanced ST SUSY. Also discussed are some three-generation GUT models found among the data sets. These models are unique because they come from basis vectors which still have a geometric interpretation -there are no "rank-cuts" in these models.