A Hybrid Matrix–Based Encoding Scheme With Error Detection via Third-Order Recurrences

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.