  The exploration of solid-solid phase transition suffers from the uncertainty
of how atoms in two crystal structures match. We devised a theoretical
framework to describe and classify crystal-structure matches (CSM). Such
description fully exploits the translational and rotational symmetries and is
independent of the choice of supercells. This is enabled by the use of the
Hermite normal form, an analog of reduced echelon form for integer matrices.
With its help, exhausting all CSMs is made possible, which goes beyond the
conventional optimization schemes. In an example study of the martensitic
transformation of steel, our enumeration algorithm finds many candidate CSMs
with lower strains than known mechanisms. Two long-sought CSMs accounting for
the most commonly observed Kurdjumov-Sachs orientation relationship and the
Nishiyama-Wassermann orientation relationship are unveiled. Given the
comprehensiveness and efficiency, our enumeration scheme provide a promising
strategy for solid-solid phase transition mechanism research.
