In this chapter, we discuss error correcting codes. We review the idea of hard versus soft decisions as types of symbol estimates that we provide to the decoder. We introduce the concepts of parity bits – which provide redundancy – and coding rate. We develop a simple, if flawed, toy systematic linear block code and use this code to demonstrate the concepts of generator matrix, hamming distance, and parity check matrix. By using the parity check matrix, we construct the syndrome and use this vector to perform error correction. To provide a set of viable systematic linear block codes, we introduce Hamming codes. We also introduce convolutional codes and relate the mathematical and shift-register block diagram forms. To enable decoding, we discuss the trellis diagram and use this diagram to motivate Viterbi decoding.

In Chapter 5, we looked at formal law-making procedures. We discussed the line that is commonly drawn between statute law, stamped with full parliamentary authority, and secondary or delegated legislation, which is made by the executive and scrutinised more or less adequately by Parliament. Occasional references were made to guidance, handbooks or codes of practice made under the authority of statute or regulation, and their use in interpreting formal statute law, notably the Data Protection Act 2018.

In the previous chapter, we tracked the path of administrative justice from a system of dispute resolution operated by courts with tribunals as spill-over machinery to a system of proportionate dispute resolution in which complaints could be resolved without resort to the formal machinery of tribunals and courts. The emphasis of that chapter was on complaining and the handling of complaints. We looked in some detail at the investigatory or ‘inquisitorial’ procedure of the ombudsman. In inquisitorial proceedings, the judge or adjudicator ‘takes full control of the proceedings, and governs the participation of the parties’ or, to put this differently, takes a central part in the proceedings with responsibility for assembling and testing the evidence with a view to eliciting the truth. The coroner’s inquest and to a certain extent the procedure followed at public inquiries, both discussed in Chapter 15, are based on a similar inquisitorial model. In this Chapter we look more closely at the evolution of what Cane calls ‘substitute tribunals’ in reference to the traditional classification of tribunals as ‘court-substitutes’. Tribunals have travelled a long way from humble beginnings to the place they occupy today as the standard machinery for alternative dispute resolution in administrative law.

Fifty years ago, contract was a low-lying feature in the administrative law landscape. This mirrored state forms, the prevalence at the time of the classic welfarist model of direct service provision by integrated, hierarchical, public bodies. It also reflected the non-development of a distinctive ‘public law’ body of legislation and jurisprudence. On the one hand, Dicey dominant, the basic premise was that government contracts should be subject to the ordinary private law; on the other, a history of Crown immunities and privileges reinforced the sense of an internal, executive-owned activity devoid of formal legal regulation. Public procurement in its traditional format of purchasing works, goods and services was both important and largely hidden from view. Behind the scenes, a well-established ‘law of the contract’ was in operation, a reservoir of standard terms and conditions on which officials could draw when specifying performance and to anticipate disputes.

Although this book does not adopt the court-centred approach of many administrative lawyers, we have learned a good deal about judicial review in its pages. Consideration of the relationship between law and administration, and the contribution law can make to administration, bears directly on the question of the proper constitutional role of the courts. Intended to produce a more rounded picture of the part played by judicial review, this chapter looks to the framing, dynamics and effects of this form of litigation. At one with the historical sense of ‘transforming judicial review’ (Chapter 4), we will encounter a changing institutional and procedural landscape, one in which different features wax and wane in importance, component parts fit together more or less well, and new elements are introduced, perhaps expansionary in effect, perhaps determinedly restrictive.

In addition to prescribing laws governing resort to force (jus ad bellum), international law also seeks to regulate the conduct of hostilities (jus in bello). These principles cover, for example, the treatment of prisoners of war, civilians in occupied territory, sick and wounded personnel, prohibited methods of warfare and human rights in situations of conflict. This subject was originally termed the laws of war and then the laws of armed conflict. More recently, it has been called international humanitarian law. Although international humanitarian law is primarily derived from a number of international conventions, some of these represent in whole or in part rules of customary international law, and it is possible to say that a number of customary international law principles exist over and above conventional rules, although international humanitarian law is one of the most highly codified parts of international law. Reliance upon relevant customary international law rules is particularly important where one or more of the states involved in a particular conflict is not a party to a pertinent convention. A good example of this relates to the work of the Eritrea–Ethiopia Claims Commission, which noted that since Eritrea did not become a party to the four Geneva Conventions of 1949 until 14 August 2000, the applicable law before that date for relevant claims was customary international humanitarian law. On the other hand, treaty provisions that cannot be said to be part of customary international law will bind only those states that are parties to them. This is particularly important with regard to some provisions deemed controversial by some states contained in Additional Protocols I and II to the Geneva Conventions 1949. One additional factor that has emerged recently has been the growing convergence between international humanitarian law and international human rights law, coupled with the relevance in this context of the rules of international criminal law. This is discussed below.

In this chapter, we discuss the concepts of baseband and passband representations of signals and the mechanisms for moving between these two forms by using up-and down-conversion. We describe multiple up- and down-conversion approaches, such as digital-only, direct, superheterodyne, and digital intermediate frequency (IF). We discuss the components used to move between baseband and passband representation: analog-to-digital and digital-to-analog converters (ADC and DAC, respectively), and frequency synthesizers.

Chapter 1 ended in a period dominated by a planned economy in which the state was a major player. The election of Margaret Thatcher as Prime Minister in 1979 marked a paradigm shift. It heralded a move from the welfarist Keynesian consensus to a market-dominated ‘New Right’ philosophy and governments dedicated to ‘rolling back the boundaries of the state’. The objective of the New Right was to restore traditional liberal values of individualism, limited government and free market forces. The buzz word was ‘freedom’. The reigning philosophers were Milton Friedman and Friedrich Hayek. Hayek, whose rule of law philosophy we met in Chapter 1, had libertarian tendencies. He believed that a rule of law state must secure as much freedom for the individual as was compatible with the freedom of others, including a measure of freedom from the state. He deprecated planning and opposed a planned economy, both socialistic features of the previous era, and thought that a ‘policy of freedom’ must minimize the harmful effects of ‘coercion’, widely defined as subjection to the forceful control of another. Friedman, an ardent exponent of monetarism, was also an ardent opponent of government, contending that, ‘while market failure justifies some state intervention … the very nature of bureaucracy and government activity makes it an undesirable alternative to market process’

When we considered cubic polynomials, we saw that a variety of possibilities can arise. Some depend on the original field 𝐾: whether or not 𝐾 contains cube roots of unity, for example. Others depend on special relationships between the coefficients: these can confuse the issue, and it is sensible to consider polynomials where this cannot happen.