Let nq(k, d) denote the smallest value ofn for which there exists a linear In, k, d]-code over GF(q). An In, k, d]-code whose length is equal to nq(k, d) is called optimal. The problem of finding nq(k, d) has received much attention for the case q = 2. We generalize several results to the case of an arbitrary prime power q as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite fidd.In particular, we study the problem with q = 3 and determine n3(k, d) for all d when k -< 4, and n3(5, d) for all but 30 values of d.

The aim of the paper is to illustrate how the housing system in the United Kingdom (UK) has contributed to creating vulnerabilities during the COVID‐19 pandemic. Drawing on the concept of ontological security we look at how living with housing insecurity whilst enduring poor housing conditions has impacted the lives of those living in households. The paper draws on semi‐structured interviews with 50 residents and 8 housing professionals. The findings outline the grinding impact of the pandemic on the ontological security of residents and the coping strategies adopted by a wider range of households who are now increasingly vulnerable. A number of people went into lockdown in vulnerable situations, experiencing deep inequalities and living in poorly maintained homes. This has weakened the ontological security experienced by many households. These represent significant failings of the housing system and housing policy impacting on the health and wellbeing of a wider cohort of people creating additional vulnerabilities.

This chapter focuses on how housing inequalities have been exacerbated by the COVID-19 pandemic. It draws on interviews with households living in poor-quality housing and investigates how pre-existing housing conditions amplify residents' experiences of that housing. It also covers physical characteristics of dwellings, such as damp and heating problems, and social aspects that include dealing with landlords and not enough space. The chapter highlights mental health challenges that are amplified by poor-quality housing, which are largely caused by people spending most of their time at home with substandard conditions that are ever-present in their lives. It highlights key findings from the new rapid empirical research undertaken within the UK, presenting the concerns, impacts, and lived experiences of both residents and professionals dealing with poor-quality housing.

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