Improved constructions of nested code pairs

Abstract

Two new constructions of linear code pairs $C_2\subset C_1$ are given for which the codimension and the relative minimum distances $M_1(C_1,C_2)$, $M_1(C_2^{\perp },C_1^{\perp })$ 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 [35]. They can also be used to ensure reliable communication over asymmetric quantum channels [47]. The new constructions result from carefully applying the Feng-Rao bounds [18,27] to a family of codes defined from multivariate polynomials and Cartesian product point sets.

36 pages

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• Improved Gilbert-Varshamov Bound for Entanglement-Assisted Asymmetric Quantum Error Correction by Symplectic Orthogonality
• On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets
• On nested code pairs from the Hermitian curve
• MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes
• Steane-Enlargement of Quantum Codes from the Hermitian Curve
• Impure codes exceeding the pure bounds for quantum local recovery