The weight hierarchy of decreasing norm-trace codes

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 sets of monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by $$x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>u</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:msup> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> over the finite field of cardinality $$q^s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:math> , where u is a positive divisor of $$\frac{q^s - 1}{q - 1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>q</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:math> . 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.