A Hybrid Matrix–Based Encoding Scheme With Error Detection via Third‐Order Recurrences
Şükran Uygun · Journal of Mathematics · 2026
In this paper, we propose a novel encoding–decoding algorithm based on third‐order linear recurrence sequences and their associated matrix representations. In particular, third‐order Jacobsthal and third‐order Jacobsthal–Lucas matrix sequences are combined with Padovan and Perrin matrix sequences to construct a hybrid block–based coding scheme. The plaintext is arranged into 3 × 3 blocks, and each block is encoded by multiplication with specially selected matrix sequences determined by a sequence tree diagram. Determinant and trace properties of the involved matrices play a fundamental role in the decoding phase. A structural performance discussion is provided analyzing encoding key domain size and algebraic robustness against representation attacks. Compared to classical Fibonacci‐based coding schemes, the use of third‐order recurrences and hybrid matrix selection increases structural complexity and enlarges the encoding key domain. An explicit example is provided to illustrate correctness and feasibility of the method.