Improved constructions of nested code pairs

TL;DR

This paper introduces two new constructions of nested linear code pairs with optimized parameters, enhancing applications in secret sharing and quantum communication by leveraging Feng-Rao bounds on multivariate polynomial codes.

Contribution

The paper presents novel nested code pair constructions with improved parameters using Feng-Rao bounds on multivariate polynomial codes.

Findings

Constructed code pairs have large codimension and relative minimum distances.

Applicable to efficient secret sharing schemes and quantum communication.

Achieves better parameter trade-offs than previous methods.

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.