Linearity of homogeneous Gray image of linear codes on Z_9

Samei, Karim; Mehri, Alireza; Eyvazi, Hamid

doi:10.22055/jamm.2026.50101.2386

Linearity of homogeneous Gray image of linear codes on Z_9

Articles in Press

Document Type : Original Paper

Authors

Karim Samei ¹

Alireza Mehri ²

Hamid Eyvazi ²

¹ Department of Mathematics Bu Ali Sina University Hamedan Iran

² Faculty of Science, Bu Ali Sina University, Iran

10.22055/jamm.2026.50101.2386

Abstract

The Z_9_additive codes are subgroups of Z_9^n of length n . Let C be a linear code of length n on Z_9 and Φ be a homogeneous Gray map on Z_9^n. In this paper, we show that the homogeneous Gray map of C is linear if and only if for every u , v in C we have u⊛v in C. When C is aperiodic and n is prime to 3, a necessary and sufficient condition for its homogeneous Gray map to be linear is given by using the generating polynomial. As a result, it is shown that the homogeneous Gray map of a square residual code on Z_9 is not linear.

Keywords

Cyclic code

homogeneous Gray map

linear code

Matheson-Salomon transform

Subjects