  We provide a theoretical framework for encoding arbitrary logical states of a
quantum bit (qubit) into a continuous-variable quantum mode through quantum
walks. Starting with a squeezed-vacuum state of the quantum mode, we show that
quantum walks of the state in phase space can generate output states that are
variants of codeword states originally put forward by Gottesman, Kitaev, and
Preskill (GKP) [Phys. Rev. A {\bf 64}, 012310 (2001)]. In particular, with a
coin-toss transformation that projects the quantum coin onto the diagonal
coin-state, we show that the resulting {\em dissipative} quantum walks can
generate qubit encoding akin to the prototypical GKP encoding. We analyze the
performance of these codewords for error corrections and find that even without
optimization our codewords outperform the GKP ones by a narrow margin. Using
the circuit representation, we provide a general architecture for the
implementation of this encoding scheme and discuss its possible realization
through circuit quantum-electrodynamics systems.

Quantum Physics