Linearity of Gray Codes via Schur Product

09/21/2023

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by   Gustavo Terra Bastos, et al.

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We propose an original approach to investigate the linearity of Gray codes obtained from ℤ_2^L-additive codes by introducing two related binary codes: the associated and concatenated. Once they are defined, one could perform a straightforward analysis of the Schur product between their codewords and determine the linearity of the respective Gray code. This work expands on earlier contributions from the literature, where the linearity was established with respect to the kernel of a code and/or operations on ℤ_2^L. The ℤ_2^L-additive codes we apply the Gray map and check the linearity are the well-known Hadamard, simplex, MacDonald, Kerdock, and Preparata codes. We also present a family of Reed-Muller codes that yield to linear Gray codes and perform a computational verification of our proposed method applied to other ℤ_2^L-additive codes.