Canal-U

4.3. Distinguisher for GRS codes

In this session we will see that generalized Reed-Solomon codes behave differently than random codes with respect to the star operation. Thus we can define a distinguisher for Generalized Reed-Solomon codes. Let us recall the definition of Generalized Reed-Solomon codes. We will need an n-tuple of mutually distinct elements of Fq. We need a vector b which is an n-tuple of nonzero elements of Fq. We need to define the vector space of all polynomials of degree at most k and we also need to define a evaluation map. Then the Generalized Reed-Solomon codes of dimension k, associated with a pair (a,b) is the evaluation of all polynomials of degree at most k at the pair (a,b). The element a is called code locator and the element b is called the column multiplier. Let us see some properties of Generalized Reed-Solomon codes. The Generalized Reed-Solomon codes is an MDS code that is its error correction performance is optimal. Moreover, the dual of a Generalized Reed-Solomon code is also a Generalized Reed-Solomon code, in particular the dual of a Generalized Reed-Solomon code of dimension k defined by the pair (a,b) is a Generalized Reed-Solomon code of dimension n-k defined by the same code locator but some non zero vector b.