On the symbol-pair distance of some classes of repeated-root constacyclic codes over Galois ring

Let $$\gamma = 4z-1$$ be an unit of Type $$(*^{-})$$ of the Galois ring $${{\,\mathrm{GR}\,}}(2^a, m)$$ . The $$\gamma$$ -constacyclic codes of length $$2^s$$ over the Galois ring $${{\,\mathrm{GR}\,}}(2^a, m)$$ are precisely the ideals $$\langle (x +1)^i \rangle$$ , $$0 \le i \le 2^sa$$ of the chain ring $$\mathfrak {R}(a,m, \gamma ) = \dfrac{{{\,\mathrm{GR}\,}}(2^a,m)[x]}{\langle {x^{2^s}} - \gamma \rangle }$$ . This structure is used to determine the symbol pair distance of $$\gamma$$ -constacyclic codes of length $$2^s$$ over $${{\,\mathrm{GR}\,}}(2^a, m)$$ . The exact symbol-pair distances for all such $$\gamma$$ -constacyclic codes of length $$2^s$$ over $${{\,\mathrm{GR}\,}}(2^a, m)$$ are obtained. Also, we provide the MDS symbol-pair codes of length $$2^s$$ over $${{\,\mathrm{GR}\,}}(2^a, m)$$ and some examples are computed.