TABLE OF CONTENTS We calls 2D binary codes or codes with two variables the codes that are ideal in 
. It is a generalization of cyclic codes that has been known for a long time.
Their algebraic structure is much more complex, because we can no longer use the Euclidian or Bezout equalities. The ring A is no longer Euclidean. It is thus difficult, except in particular cases, to know the performances of these codes well.
Any word of a 2D code can be represented as a polynomial with two variables c( X, Y ). A 2D code can have a system of generators not reduced to a polynomial, contrary to cyclic codes. The results of Gr bner and B. Buchberger are quite useful to process these codes.
A very simple case of 2D codes is that of product codes. A product code is an ideal of A generated by the generator g( X, Y) = g 1( X) g 2( Y), where g 1( X) divides X n ? 1 and g 2 ( Y) divides Y m ? 1. This code is the set of the multiples of g( X, Y) in A.
We have the two following results:
Its dimension is k 1 k 2, where k 1 is the dimension of the code 
and k 2
