Chapter 2 is all preliminary material. Section 2.1 reviewsp-adic fields and Galois rings, and Sections 2.2 and 2.3 review the Fourier transform. Section 2.4 introduces the various weight functions we shall consider, and introduces the notion of normalized weight, which is our device for dealing with codes in which 1A is in the support of the Fourier transform (i.e., codes with constant words). Of particular note is Section 2.5, which introduces the notion of accounts, which are simply functions from a setY into the integers. Accounts that take nonnegative values are regarded as multisets. This combinatorial device is indispensable for making our equations (barely) compact enough to display conveniently on the page. Any reader who wishes to understand the calculations performed here must be familiar with the notations for accounts. Sections 2.6 and 2.7 include various combinatorial devices that we employ to obtain our more precise results (such as proofs of sharpness). We recommended that the reader pass over these sections and return to them only when the tools they describe are actually employed (in parts of Chapters 4–7). Section 2.8 describes some notations we use for multivariable polynomials as well as some basic facts about polynomials that we shall need. The reader should be familiar with the notations set down there because they are often used.
Chapter 3 provides abstract theorems that will give p-adic estimates of weights if one can furnish an appropriate counting polynomial. Nothing is said about how to find counting polynomials; this will be done in each of the succeeding chapters as the need arises.
In Chapter 4, we prove Theorems 1.8 and 1.7, while laying down the foundations needed for all our counting polynomial constructions. Section 4.1 has some fundamental material on the Newton expansion, which is the mathematical device underlying all our counting polynomial constructions. We construct our counting polynomials in Section 4.2, some of which will be used in Chapters 6 and 7. We show how to employ the polynomials in Section 4.3. The techniques and results of Sections 4.1–4.3 will be reused in Chapter 5. In Sections 4.4 and 4.5, we use our counting polynomials to prove Theorems 1.8 and 1.7, and in Section
4.6 we compare these results with earlier work.
Chapter 5 is dedicated to proving Theorem 1.9. After discussing previous results and comparing our new theorem with them (relying heavily on material from Section 4.6 for the comparison), we spend the next two sections (5.1 and 5.2) specializing the notions of Sections 4.2 and 4.3 for Lee weight. Thus we obtain a counting polynomial, which we use in Section 5.3 to prove Theorem 1.9.
Chapter 6 is dedicated to proving Theorem 1.10. In Sections 6.1–6.3, we develop the trace-averaging procedure to obtain an appropriate counting polynomial. We derive Theo- rem 1.10 in Section 6.4 and show that we can recover some earlier results from this highly general theorem in Section 6.5.
In Chapter 7, we prove Theorem 1.11 and relate it to the theorem of N. M. Katz (Theorem 1.12). In Sections 7.1–7.3, we carry out a more specialized version (for finite fields only) of the trace-averaging procedure of Chapter 6. Then in Section 7.4, we construct our polynomial for counting simultaneous zeroes. We use this polynomial in Section 7.5 to prove Theorem 1.11. In Section 7.6, we review the theorems of Chevalley-Warning, Ax, and N. M. Katz. In Section 7.7, we show how to translate results about weights in group algebras to results about cardinalities of affine algebraic sets over finite fields. In Section 7.8, we prove the theorem of N. M. Katz and the associated statement about its sharpness.
Chapter 2
Preliminaries
In this chapter, we review the fundamental mathematics needed to state and prove our results. We also introduce definitions, notations, and combinatorial devices that allow us to describe and manipulate the objects that arise in this study. We discuss p-adic fields and Galois rings in Section 2.1. Then we introduce the Fourier transform for the group algebra R[A] (with R a fairly generic ring) in Section 2.2. In Section 2.3, we give more specialized results on the Fourier transform in the case when R is a Galois ring or a ring of integers in an unramified extension of the p-adics. In Section 2.4, we review weight functions commonly used in algebraic coding theory. There we introduce the normalized weight function, a device for simplifying the presentation of our results when our code has the trivial character in the support of its Fourier transform (i.e., when our code contains constant words).
The second half of this chapter deals more with notations and devices that make it easier for us to state and prove our theorems. Section 2.5 is especially critical in this regard. There we introduce the notion of an account (which is a generalization of a multiset) and tools for manipulation of accounts. These accounts are ubiquitous in this thesis, so the reader must be familiar with them, with their notation, and with the basic operations that can be performed upon them. Section 2.6 introduces the procedures called collapse and reduction of accounts. These devices are needed in our proofs of sharpness of lower bounds on p- adic valuations of weights. The reader should probably skip this section until collapse and reduction are actually used (beginning in Section 4.4). Section 2.7 introduces theFrobenius action on accounts. This device is used to prove that certain quantities in our p-adic
estimates, which we already know to be elements ofZp[ζq−1], are in fact elements ofZp. It would be a good idea for readers to skip this section, and return to it only if they want to understand the use of the Frobenius action in the proofs of Theorems 4.18, 5.12, 6.13, and 7.14. Section 2.8 includes notations we use with multivariable polynomials. There we also include a basic fact about polynomials, which we use to prove that certain of our bounds on p-adic valuations of weights are sharp.