TL;DR
This paper introduces a new class of algebraic codes derived from scrolls over algebraic curves, generalizing Goppa codes, with a novel decoding approach based on vector bundle extensions.
Contribution
It defines scroll codes over curves of any genus, explores their properties, and links decoding to vector bundle stability, extending the theory of algebraic geometric codes.
Findings
Scroll codes generalize Goppa codes over finite fields.
Decoding involves identifying destabilizing subbundles.
Orthogonal complements of these codes also belong to the same class.
Abstract
We define and study a class of codes obtained from scrolls over curves of any genus over finite fields. These codes generalize Goppa codes in a natural way, and the orthogonal complements of these codes belong to the same class. We show how syndromes of error vectors correspond to certain vector bundle extensions, and how decoding is associated to finding destabilizing subbundles.
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