Improved Constructions of Nested Code Pairs.

Two new constructions of linear code pairs C2 \subset C1 are given for which the codimension and the relative minimum distances M1(C1,C2) and M1(C\vphantom {Rj2}^\perp,C1^\perp) are good. By this, we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes. They can also be used to ensure reliable communication over asymmetric quantum channels. The new constructions result from carefully applying the Feng–Rao bounds to a family of codes defined from multivariate polynomials and Cartesian product point sets. [ABSTRACT FROM AUTHOR]