Generalized Reed-Muller codes: A new construction of information sets
José Joaquín Bernal
TL;DR
The paper develops a defining-set–based framework to construct information sets for q-ary Generalized Reed-Muller codes of first and second order by mapping punctured cyclic GRM codes to two-dimensional abelian codes and exploiting the structure of their defining sets. For first-order GRM codes, the resulting information sets coincide with the classical RM constructions, while for second-order GRM codes with $q>2$ they yield new, more intricate information sets requiring a refined analysis of $q$-cyclotomic cosets and restricted representatives. The authors establish the main theoretical machinery by combining two-dimensional abelian-code results with GRM duals, and illustrate the method with concrete examples, including an explicit first-order case and a second-order example with $q=5$. Overall, the work advances the practical construction of information sets for GRM codes and opens avenues for higher-dimensional generalizations and decoder adaptations, such as permutation decoding, in the q-ary setting.
Abstract
In [2] we show how to construct information sets for Reed-Muller codes only in terms of their basic parameters. In this work we deal with the corresponding problem for q-ary Generalized Reed-Muller codes of first and second order. We see that for first-order codes the result for binary Reed-Muller codes is also valid, while for second-order codes, with q > 2, we have to manage more complex defining sets and we show that we get different information sets. We also present some examples and associated open problems.