Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric

• Published in: IEEE transactions on information theory Vol. 66; no. 7; pp. 4061 - 4074
• Main Authors: Santos, Welington, Alves, Marcelo Muniz
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-07-2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Binary codes; Binary system; Codes; Extraterrestrial measurements; invariant theory; Invariants; Linear codes; Metric space; Orbits; ordered hamming spaces; Polynomials; poset codes; Self-dual codes; Shape; shape enumerator; Space vehicles
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.2971989

In this paper we consider self-dual NRT codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman metric (NRT metric) and their shape enumerators as defined by Barg and Park. We use polynomial invariant theory to describe the shape enumerator of a binary self-dual NRT code, even self-dual NRT code, and weak doubly even self-dual NRT code in <inline-formula> <tex-math notation="LaTeX"> {M}_{ {n},2}(\mathbb {F}_{2}) </tex-math></inline-formula>. Motivated by these results, we describe the number of invariant polynomials that we must find to describe the shape enumerator of a self-dual NRT code in <inline-formula> <tex-math notation="LaTeX"> {M}_{ {n}, {s}}(\mathbb {F}_{2}) </tex-math></inline-formula>. We define the ordered flip of a matrix <inline-formula> <tex-math notation="LaTeX"> {A}\in {M}_{ {k},{ { ns}}}(\mathbb {F}_{ {q}}) </tex-math></inline-formula> and present some constructions of self-dual NRT codes over <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{ {q}} </tex-math></inline-formula>. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two.