SAT to k-in-3-SAT reduction

Question

Given a 3-SAT clause. Is there a way to convert 3-SAT to k-in-3-SAT such that:

The number of new variables introduced are less than the number of clauses (without adding dummy clauses etc.)?

The standard way is by introducing 4 new variables and 3 new clauses as mentioned here: https://en.wikipedia.org/wiki/Boolean_satisfiability_problem

If not is it possible for some N-SAT (N>3)?

Answer 1

There is a way to convert 2SAT to X3SAT that adds only one new variable per clause.

$(a \lor b) \rightarrow (\bar a, \bar b, x)$

where $x$ is a new variable unique to this clause.

I don't know of any shorter methods for converting 3SAT other than the normal method.