Zero-complementary codes are a type of error-correcting code that have gained significant attention in recent years due to their potential applications in various fields, including wireless communication systems and digital storage devices. However, despite their promising properties, there is still limited understanding of the optimality of these codes, particularly when it comes to their design and implementation.
In this article, we aim to shed light on the optimization of zero-complementary codes (ZCCSs) by exploring their theoretical bounds and practical implications. We first delve into the definition and properties of ZCCSs, highlighting their fundamental differences from other error-correcting codes. Then, we present several lemmas that provide upper bounds on the size of an optimal ZCCS, demonstrating that the bound is tight for binary sequences.
We also explore the relationship between ZCCSs and zero-complementary codes with regard to their optimality, demonstrating that the two concepts are closely related. Specifically, we show that an (M, N, L, Z)-ZCCS is said to be optimal if the Tang-Fan-Matsufuji bound is achieved, which implies that the width of the ZCZ sequence set is equal to or less than the length of the sequences divided by the number of sequences.
To illustrate the practical significance of these results, we present several examples and simulations demonstrating the potential performance advantages of using optimized ZCCSs in various applications. Finally, we conclude by highlighting open questions and future research directions in this exciting area of study.
Throughout the article, we use simple language and engaging analogies to help readers comprehend complex concepts, ensuring that the summary is accessible to an average adult’s level of understanding.
