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We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\log n)^{-2/3})$.
The Classical house can be understood also as a set of interactions between people, objects and valuable commodities that existed within and extended beyond the physical and temporal confines of the house. This chapter concerns ceramic oil jars, lekythoi, that date to the early 5th century BCE. It is argued that the small black-figured lekythoi, which were prolifically produced and widely traded during this time, may have held olive oil, not just perfume. The materiality of these pots and archaeological evidence from settlements, graves and other find-spots suggest that lekythoi could have functioned as oil pitchers to serve small portions of olive oil, perhaps of family production. Vase iconography indicates that such lekythoi were objects within easy reach, to be used on diverse occasions, such as dining, ritual and commercial activities. The offering of lekythoi in burials, irrespective of the presence of contents, could have alluded to the storage of olive oil in the household of the deceased and communicated a powerful message about a family’s claims to status, real or fictitious.
Books and boxes were found in close proximity; before bookshelves, chests were the most obvious place to store books, and the physical features of a bound book often made it visually analogous to a box. The material and tactile connections between book and box play into common metaphors of the book as a receptacle for textual riches, and the chapter brings together responses to the book as box-like object from Erasmus to early seventeenth-century English Protestants, from humanist treatises to portraits. In considering literary and visual encounters with the codex, discussion focuses on the significance of external surfaces, such as gold, blackness, and embroidery, in the fashioning of these inherently box-like objects. While reformers insisted on the Word of God as the only vehicle of truth, they could not escape the fact that it had to be contained in books, unavoidably material receptacles with insides and outsides that could shape and inscribe each other. Protestants remained sensitive to the metaphorical potential of an object with insides and outsides, and this chapter demonstrates that the identity of the ‘book’ was more complex than 'sola scriptura' suggested.
Focuses on the reliquary: an enclosing, revealing structure, which engages intensely with its contents. The apparently idolatrous worshipping of ‘Gods bodye in the box’ was a persistent complaint among reformers, and the murky box of the reliquary epitomised the falseness of the Roman Catholic faith. The chapter starts with sixteenth-century encounters with relics, beginning with Erasmus, whose attitude is characterised by ambiguities about the spiritual significance of material things. Comparing satirical and polemical responses to relics from both sides of the religious divide, the chapter considers how these boxes operated as contested sites. It then turns to the afterlife of the reliquary once it had been removed from the religious sphere, and locates its survival in the vocabulary of post-Reformation libraries as new kinds of shrine, and in seventeenth-century printed reliquiae, as safer kinds of receptacle. Even after the reliquary appeared to be emptied of its dangerous significance, the very idea of the relic and the possibilities offered by this controversial box endured as ways of thinking about the interweaving of physical and intellectual apprehension demanded by books.
Efficient seed storage is a shared concern among the growing number of seed banks established for crop improvement or ex situ conservation. Container properties greatly affect seed interactions with the environment and the overall cost and success of seed banking operations. Several material properties contribute to their suitability as seed containers. This paper provides a consolidated list of water vapour permeability properties of thermal plastics commonly used for packaging. Composite packages with layers of film with different properties provide distinct advantages to seed banks. Different seed banks must rank the importance of the various factors depending on their mission and resources. Once the risks, costs and benefits are weighed, an appropriate strategy can be developed that addresses a seed bank's specific needs. Because there are many problems and several solutions, it is likely that strategies will vary among seed banks. This response details variables to consider when selecting seed storage packages, and focuses on water diffusion rates of packages with different compositions. A ‘moisture audit’ will help seed bank operators make informed decisions about packaging.