construct error correcting codes, e. g. [@LMSS] or [@RU], motivating the $\mod 3-$case. A very recent study of the $\mod 2-$ case is [@ACHLIO]. More literature can be found in [@KO], but precise threshold results have not been obtained.

Contents
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  I. Equations modulo $3$                                                      
  1\. Notation and basics                                                      
  2\. Outline of the proof of Theorem \[EX2EI\]                                
  3\. Proof of Theorem \[OPT\]                                                 
  3.1 Proof of Lemma \[lem1\]                                                  
  3.2 Proof of Lemma \[lem2\]                                                  
  3.3 Proof of Lemma \[lem3\]                                                  
  3.4 Proof of Lemma \[lem4\]                                                  
  4\. Proof of Theorem \[LAPLA\]                                               
  5\. Remaining proofs                                                         
  5.1 Local limit consideration                                                
  5.2 The sharp threshold                                                      
                                                                               
  II\. Uniquely extendible constraints                                         
  1\. Outline                                                                  
  2\. Proof of Theorem \[UNOPT\] for $d=4,\, s\ge 7 , \,  \lambda \le 1-1/d$   
  3\. Proof of Theorem \[UNOPT\] for $d=4,\, \lambda \ge 1-1/d, s \ge 5.$      
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[**I. Equations modulo $3$** ]{}

Notation and basics
===================

We use the abbreviation $$\begin{aligned}
M(m, n) \,:= \, \sum_{ v_1, \dots v_n \,\ge 2}\, {m \choose v_1,\dots v_n } \mbox{ and } 
N_0 \,:= \, M(km, n). \mbox{ Then } N_0\cdot 3^m  \label{DEFN0} 
 \end{aligned}$$ is the number of all formulas with $ k $ variables per equation and $ m $ equations. We 