The weight hierarchy of decreasing norm-trace codes

TL;DR

This paper determines the generalized Hamming weights of decreasing norm-trace codes, extending known results to new code families and applying findings to quantum code construction.

Contribution

It introduces the weight hierarchy of decreasing norm-trace codes and relates it to one-point norm-trace and Hermitian codes, also exploring quantum code applications.

Findings

Generalized Hamming weights of decreasing norm-trace codes are explicitly determined.

The weight hierarchy of one-point norm-trace codes is recovered as a special case.

Relative generalized Hamming weights are used to construct quantum codes with excellent parameters.

Abstract

The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, $t$-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by $x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$ over the finite field of cardinality $q^s$, where $u$ is a positive divisor of $\frac{q^s - 1}{q - 1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters.