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In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing some sequences that can potentially construct all GS sequences. Firstly, it is proven that a quad of ± 1 polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to { 0 , ± 1 } . Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and ± 1 . One more motivation is to obtain these sequences more efficiently. To this end, we make use of the k-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences. [ABSTRACT FROM AUTHOR]

This correction notice is for an article titled "Sodium, potassium intake, and all-cause mortality: confusion and new findings" published in BMC Public Health. The correction addresses an incorrect author name, with the correct name being Ping Zeng instead of Ping Zen. The original article has been updated to reflect this correction. The publisher, Springer Nature, remains neutral in terms of jurisdictional claims and institutional affiliations. The authors of the article are Donghao Liu, Yuqing Tian, Rui Wang, Tianyue Zhang, Shuhui Shen, Ping Zeng, and Tong Zou. [Extracted from the article]