Consider a simple, connected graph Γ withnvertices. LetCbe a code of lengthnwith its coordinates corresponding to the vertices of Γ. We defineCas astorage codeon Γ if, for any codewordc∈C, the information at each coordinate ofccan be recovered by accessing its neighboring coordinates. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we employ the polynomial method to address a question proposed by Barg and Zémor in 2022, demonstrating that the BCH family of storage codes on triangle-free Cayley graphs achieves a unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs. We also compare the BCH family with the other known constructions by examining the rate of convergence of 1/(1-R(Cn)) with respect to the lengthn, whereR(Cn) is the rate of codeCn. At last, we reveal a connection between the storage codes on triangle-free graphs and the Ramsey numberR(3,t), which leads to an upper bound for the rate of convergence of 1/(1 -R(Cn)).

Let G be a finite abelian group. If f : G → C is a nonzero function with Fourier transform f, the Donoho-Stark uncertainty principle states that |supp(f)||supp(f̂)| ≥ |G|. The purpose of this paper is twofold. First, we present the shift bound for abelian codes with a streamlined proof. Second, we use the shifting technique to prove a generalization and a sharpening of the Donoho-Stark uncertainty principle. In particular, if f : G → F is a non-zero function from G to a field F, and if f has a Fourier transform f̂, the sharpened uncertainty principle states that |supp(f)||supp(f̂)| ≥ |G|+|supp(f)|-|H(supp(f))|, where H(supp(f)) is the stabilizer of supp(f) in G.

MOTIVATION: Cell-based phenotypic screens using small molecule inhibitors is an important technology for early drug discovery if the relationship between the disease-related cellular phenotype and inhibitors' biological targets can be determined. However, chemical inhibitors are rightfully believed to be less specific than perturbation by biological agents, such as antibody and small inference RNA. Therefore, it is often a challenge in small molecule phenotypic screening to infer the causality between a particular cellular phenotype and the inactivation of the responsible protein due to the off-target effect of the inhibitors. RESULTS: In this article, we present a Roche in-house effort of screening 746 structurally diverse compounds for their cytotoxicity in HeLa cells measured by high content imaging technology. These compounds were also systematically profiled for the targeted and off-target binding affinity to a panel of 25 pre-selected protein kinases in a cell-free system. In an effort to search for the kinases whose activities are crucial for cell survival, we found that the simple association method such as the chi-square test yields a large number of false positives because the observed cytotoxic phenotype is likely to be the result of promiscuous action of less specific inhibitors instead of true consequence of inactivation of single relevant target. We demonstrated that a stratified categorical data analysis technique such as the Cochran-Mantel-Haenszel test is an effective approach to extract the meaningful biological connection from the spurious correlation resulted from confounding covariates. This study indicates that, empowered by appropriate statistical adjustment, small molecule inhibitor perturbation remains a powerful tool to pin down the relevant biomarker for drug safety and efficacy research. CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

In this paper, we study ternary monomial functions of the formf(x) = Trn(axd), wherexisin \BBF3nandTrn: \BBF3nrarr \BBF3is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising in the 2006IEEE Transactions on Information Theorypaper (vol. 52, pp. 2018-2032, 2006) are weakly regular bent, thus settling a conjecture of Helleseth and Kholosha. We also prove that the Coulter-Matthews bent functions are weakly regular.

An explicit construction of a family of binary low-density parity check (LDPC) codes called$hbox LU(3,q)$, where$q$is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when$q$is odd. The conjecture is proved in this note. The proof involves the geometry of a four-dimensional (4-D) symplectic vector space and the action of the symplectic group and its subgroups.

We give direct and recursive constructions for aperiodic and periodic complementary sequences. Using these constructions, many missing entries in the table of Bomer and Antweiler (1990) can be filled.

Recently, Maschietti (Des., Codes Cryptogr., vol.14, p.89, 1998) constructed several families of (2/sup m/-1,2/sup m-1/-1,2/sup m-2/-1) cyclic difference sets from monomial hyperovals. In this correspondence, we consider the binary sequences associated to the difference sets constructed by Maschietti. We give algebraic proof of the fact that these sequences have two-level autocorrelation functions, also we discuss the linear spans of these sequences.