Summary: We introduce a new Fibonacci \(G_{p,m}\) matrix for the \(m\)-extension of the Fibonacci \(p\)-numbers where \(p\) \((\geq 0)\) is an integer and \(m\) \((>0)\). Thereby, we discuss various properties of the \(G_{p,m}\) matrix and the coding theory that followed from the \(G_{p,m}\) matrix. In this paper, we establish the relations among the code elements for all values of \(p\) (nonnegative integer) and \(m\) \((>0)\). We also show that the relation among the code matrix elements for all values of \(p\) and \(m=1\) coincides with the relation among the code matrix elements for all values of \(p\) [the authors, Chaos Solitons Fractals 41, No. 5, 2517–2525 (2009; Zbl 1198.94196)]. In general, correctability of the method increases as \(p\) increases but it is independent of \(m\).
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