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[Formula: see text]A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [Formula: see text], we find the largest Reed–Muller code contained in [Formula: see text] and the smallest Reed–Muller code containing [Formula: see text]. We then prove that similar to Reed–Muller and affine Cartesian codes, the [Formula: see text]th generalized Hamming weight and the [Formula: see text]th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the [Formula: see text]th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the [Formula: see text]th footprint of a hyperbolic code that, sometimes, are sharp.
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Algebraic hierarchical locally recoverable codes with nested affine subspace recovery
Abstract Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.
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- Award ID(s):
- 2201075
- PAR ID:
- 10550453
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Designs, Codes and Cryptography
- Volume:
- 93
- Issue:
- 1
- ISSN:
- 0925-1022
- Format(s):
- Medium: X Size: p. 111-132
- Size(s):
- p. 111-132
- Sponsoring Org:
- National Science Foundation
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