Multiplicity codes reveal their symmetry groups under nrt metric
The permutation and automorphism groups of univariate multiplicity codes
Information Theory
Summary
The paper finds exactly which ways you can rearrange or transform certain special error-correcting codes, called univariate multiplicity codes, without changing their essential structure. These codes are studied with respect to a particular way of measuring distance called the Niederreiter-Rosenblum-Tsfasman metric. The authors identify the precise symmetries and permutations for any code size, multiplicity, and choice of points to evaluate at. For simpler codes like classical Reed-Solomon codes, familiar symmetry groups are recovered, but for more complex multiplicity levels, the symmetries become much more restricted.
What this means in practice
- •For error correction engineers: Determine the exact symmetries of multiplicity codes to optimize encoding and decoding algorithms under the NRT metric.
- •For network designers: Use knowledge of code automorphisms to design robust communication schemes that leverage multiplicity codes with controlled symmetry properties.
A theory result. No direct application yet.
Authors
Eduardo Camps Moreno, Hiram H. López, Umberto Martínez-Peñas, Rodrigo San-José
Abstract
We give explicit characterizations of the permutations and linear isometries that leave (univariate) multiplicity codes invariant. We consider any evaluation subset of the finite field, any multiplicity and dimension, and the automorphisms are with respect to the Niederreiter-Rosenblom-Tsfasman (NRT) metric. For multiplicity one, we recover the permutation and automorphism groups of classical Reed--Solomon codes in the classical Hamming metric. In the main dimension range, automorphisms are uniquely determined by a projectivity that permutes the evaluation subset, up to multiplication by a nonzero scalar. For multiplicity one, permutations are given by affine maps that permute the evaluation subset. In contrast, for multiplicity at least two, permutations are only given by translations that permute the evaluation subset. Finally, we give an algebraic characterization of automorphisms of multiplicity codes for the folded Hamming metric.