Matrix representation, determinants, and inverses of Gaussian Lehmer sequences for biperiodic error-correcting codes

This paper introduces the Gaussian Lehmer sequence and presents its representation using a lower triangular Pascal matrix. Building on this matrix form, a new coding method is developed with improved capabilities for error detection and correction. The study explores the use of this coding approach in the realm of complex numbers within coding theory. Further investigation shows that the sequence follows distinct recurrence relations for odd and even terms. To unify these behaviors, a new function is proposed that captures both cases, resulting in the definition of a biperiodic sequence. This paper also provides a thorough analysis of the circulant matrix generated by the sequence, including the computation of its determinant and inverse, highlighting the promise of this matrix-based framework in advancing coding theory.