Charcoal, 30 29 28 bytes

ＮθＮηＩＥＸθηＥ÷ιＸθ⮌…⁰η⌊﹪⟦⊖±λλ⟧⊗θ

Try it online! Link is to verbose version of code. Outputs each digit of the gray code as a base 10 number on its own line, with each gray code double-spaced from the next. Explanation:

Ｎθ                              Input `N` as a number
  Ｎη                            Input `k` as a number
       θ                        Input `N`
      Ｘ                         To the power of
        η                       Input `k`
     Ｅ                          Map over implicit range
           ι                    Current value
          ÷                     Vectorised integer divide
             θ                  Input `N`
            Ｘ                   Vectorised to power
               …                Range from
                ⁰               Literal integer `0`
                 η              To input `k`
              ⮌                 Reversed
         Ｅ                      Map over list
                       λ        Current value
                      ±         Negated
                     ⊖          Decremented
                    ⟦    ⟧      List of that and
                        λ       Current value
                   ﹪            Vectorised modulo
                           θ    Input `N`
                          ⊗     Doubled
                  ⌊             Take the minimum
    Ｉ                           Cast to string
                                Implicitly print