Abstract
In this paper, we introduce a novel class of real number codes termed as [Formula: see text]-spectral codes, derived from real symmetric operators. Our focus is on addressing the challenges associated with error correction in communication systems utilizing real number codes. Our study initiates with a meticulous definition and analysis of the fundamental properties of [Formula: see text]-spectral codes. These properties serve as the foundation for comprehending the distinctive characteristics that [Formula: see text]-spectral codes offer, ensuring the integrity and reliability of transmitted data. Our exploration extends to the construction of maximum distance separable (MDS) [Formula: see text]-spectral codes, which boast the maximum error correction capability. As a culmination of our research, we present an efficient decoding technique tailored for [Formula: see text]-spectral codes. Leveraging an MDS [Formula: see text]-spectral code and our decoding technique, we achieve optimal error correction and ensure good numerical stability. This innovative approach opens new avenues for enhancing the performance of real-number codes in practical applications.
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