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Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order $v$ is partitioned into two pairs. Each pair in such a partition is called a nested design pair and its multiplicity is the number of times it is a pair in this partition. Such a partition of each block is considered as a new block design called a nested Steiner quadruple system. Several related questions on this type of design are considered in this paper: What is the maximum multiplicity of the nested design pair with minimum multiplicity? What is the minimum multiplicity of the nested design pair with maximum multiplicity? Are there nested quadruple systems in which all the nested design pairs have the same multiplicity? Of special interest are nested quadruple systems in which all the $\binom{v}{2}$ pairs are nested design pairs with the same multiplicity. Several constructions of nested quadruple systems are considered and in particular classic constructions of SQS are examined.

Ethereum, as a representative of Web3, adopts a novel framework called Proposer Builder Separation (PBS) to prevent the centralization of block profits in the hands of institutional Ethereum stakers. Introducing builders to generate blocks based on public transactions, PBS aims to ensure that block profits are distributed among all stakers. Through the auction among builders, only one will win the block in each slot. Ideally, the equilibrium strategy of builders under public information would lead them to bid all block profits. However, builders are now capable of extracting profits from private order flows. In this paper, we explore the effect of PBS with private order flows. Specifically, we propose the asymmetry auction model of MEV-Boost auction. Moreover, we conduct empirical study on Ethereum blocks from January 2023 to May 2024. Our analysis indicates that private order flows contribute to 54.59% of the block value, indicating that different builders will build blocks with different valuations. Interestingly, we find that builders with more private order flows (i.e., higher block valuations) are more likely to win the block, while retain larger proportion of profits. In return, such builders will further attract more private order flows, resulting in a monopolistic market gradually. Our findings reveal that PBS in current stage is unable to balance the profit distribution, which just transits the centralization of block profits from institutional stakers to the monopolistic builder.

To measure repair latency at helper nodes, we introduce a new metric called skip cost that quantifies the number of contiguous sections accessed on a disk. We provide explicit constructions of zigzag codes and fractional repetition codes that incur zero skip cost

Locally repairable codes (LRCs), which can recover any symbol of a codeword by reading only a small number of other symbols, have been widely used in real-world distributed storage systems, such as Microsoft Azure Storage and Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding and decoding complexity, constructions of binary LRCs are of particular interest. The aim of this paper is to construct dimensional optimal binary locally repairable codes with disjoint local repair groups. We introduce how to connect intersection subspaces with binary locally repairable codes and construct dimensional optimal binary linear LRCs with locality $2^b$ ($b\geq 3$) and minimum distance $d\geq 6$ by employing intersection subspaces deduced from the direct sum. This method will sufficiently increase the number of possible repair groups of dimensional optimal LRCs, and thus efficiently expanding the range of the construction parameters while keeping the largest code rates compared with all known binary linear LRCs with minimum distance $d\geq 6$ and locality $2^b$ ($b\geq 3$)., Comment: Accepted for publication in the SCIENCE CHINA Information Sciences