A new class of variable-length, and statistically-synchronisable codes is presented in conjunction with code-construction methods and some significant property theorems. A notable feature of the codes (augmented T-codes) is their ability to recover correct character synchronisation under normal decoding procedures, following timing or additive errors in a serial-message sequence. As a result of the code construction, the synchronisation delay is found to be largely independent of the source, with the expected delay typically fewer than three characters. A further class of associated fixed-length binary codes (the binary-depletion T-codes) is introduced, and shown to be important in systematic applications for code generation, encoding/decoding operations and determining synchronisation status in digital processors.