The homogeneous Gray image of linear codes over the Galois ring \(\mathrm{GR}(4, m)\) (Q6571304)

The Galois ring \(R=\mathrm{GR}(4, m)\) is a ring of characteristic 4 and cardinality \(4^m.\) A linear code is called cyclic, if a cyclic shift of any codeword is a codeword, and a code of length \(n\) is linear over \(R\) if it is a \(R\)-submodule of \(R^n.\) If \(u, v \in\mathbb{Z}_4,\) \([u] = [u_0, u_1]\) and \([v] = [v_0, v_1],\) then \(u \odot v = u_0v_0 + 2u_1v_1. \) The natural epimorphism \(\overline{\phantom{x}}:R[x]\to \mathbb{F}_{2^m}\) is defined as \(a(x) + 2b(x) \mapsto a(x).\) For a cyclic code of odd length \(n,\) there is a unique set of pairwaise coprime monic polynomials \(g_0, g_1, g_2\) over \(R,\) such that \(g_0g_2g_1 = x^n-1\) in \(R[x],\) and \(\mathcal{C} =\langle g_0g_1, 2g_1g_2 \rangle.\)\N\NFor a linear code \(\mathcal{C}\) of length \(n\) over \(R\) and let \(\Phi\) a Homogeneous Gray map on \(R^n,\) the authors are interested in under what conditions the image of this code is linear or non-linear. It is shown that the code \(\Phi(\mathcal{C})\) is linear if and only if for every \(X, Y \in\mathcal{C},\) it follows that \(2(X \odot Y)\in\mathcal{C}.\)\N\NUsing the generator polynomial of a cyclic code of odd length over \(R,\) the necessary and sufficient condition of \(\langle \overline{g_0}\,\overline{g_1}\rangle\ast \langle \overline{g_0}\,\overline{g_1}\rangle\subset\langle \overline{g_0}\rangle\) is given for its Gray image to also be linear. Examples of linear codes over \(\mathrm{GR}(4, 2)\) of lengths 6 and 7, as well as cyclic codes of lengths 3 and 7, are given.