On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple (X,P,E) from the algebraic geometry code C=CL(X,P,E) , where X is an algebraic curve over the finite field Fq,P is an n-tuple of Fq -rational points on X and E is a divisor on X . If deg(E)=2g+1 where g is the genus of X , then there is an embedding of X onto Y in the projective space of the linear series of the divisor E. Moreover, if deg(E)=2g+2 , then I(Y) , the vanishing ideal of Y , is generated by I2(Y) , the homogeneous elements of degree two in I(Y) . If n>2deg(E) , then I2(Y)=I2(Q) , where Q is the image of P under the map from X to Y . These three results imply that, if 2g+2=m