A Z2s -additive code C of length n is a subgroup of Zn2s . Carlet (IEEE Trans Inf Theory 44:1543–1547, 1998) introduced a Gray map Φ (Carlet’s Gray map) on Z2s and Tapia-Recillas and Vega (SIAM J Discret Math 17:103–113, 2003) have been shown that Φ(C) is a binary linear code if and only if for every u,v∈C , 2(u⊙v)∈C , which their number is ∣C∣2−(∣C∣2) . Let C be a linear code over Z8 of type {δ0,δ1,δ2} . In this paper, by using the rows of the generator matrix (in standard form) of C , we introduce a set S with ∣S∣≤δ0(δ0+3)2+δ1 and show that Φ(C) is linear if and only if for every u,v∈S , 2(u⊙v)∈C . The results show that for a linear code C over Z8 calculations to check the linearity of Φ(C) are less than Tapia-Recillas and Vega’s method. In addition, for a cyclic code of odd length over Z8 , a condition on the generator polynomial of this code is given which its Carlet’s Gray image is linear. Also, as a result, we show that the Carlet’s Gray image of a quadratic residue code over Z8 is not linear.
