Existence of non-isomorphic permutation-free codes

Establish that permutation-free decimal length-3 check-digit codes exist that are non-isomorphic to the code in Table III, while retaining the relevant error-detecting properties.

Background

The paper constructs a permutation-free decimal length-3 check-digit code in Table III and shows that it belongs to a group of six conjugate codes obtained by permuting the positions of codeword triplets. These conjugate codes share the same error-detecting properties and are pairwise disjoint except for the triple codewords included in each.

After describing this family, the paper states a conjecture that additional codes exist beyond the conjugate constructions. The meaning of “isomorphic” is not specified in the paper; the author explicitly leaves that interpretation to the reader, so the unresolved issue is the existence of genuinely non-isomorphic codes under an appropriate notion of equivalence.