# Algebraic Geometry Codes: Advanced Chapters

###### Michael Tsfasman, Serge Vlǎduţ, Dmitry Nogin
Publisher:
AMS
Publication Date:
2019
Number of Pages:
453
Format:
Hardcover
Series:
Mathematical Surveys and Monographs
Price:
129.00
ISBN:
978-1-4704-4865-3
Category:
Monograph
[Reviewed by
Nathan Kaplan
, on
08/1/2020
]
Algebraic Geometry Codes: Advanced Chapters is a sequel to an earlier book by the same so I will start this review by recalling just a small amount about where that book left off and this one begins.

A major focus of Basic Notions is the study of linear codes that come from algebraic curves. Let $X$ be a projective, smooth, absolutely irreducible algebraic curve defined over a finite field $\mathbb{F}_{q}$. Let $\mathcal{P}= \left\{ P_{1}, P_{2}, \ldots P_{n} \right\}$ be a subset of the $\mathbb{F}_{q}$ -rational points of $X$. Let $D$ be a divisor on $X$ whose support does not include any of the $P_{1}$ and let  $L(D)$ denote the Riemann-Roch space of $D$. Consider the evaluation map

$\operatorname{Ev}_{\mathcal{P}} \colon L(D) \to \mathbb{F}_{q}^{n}, \;\; f \to \left( f(P_{1}), \ldots, f(P_{n}) \right)$

The image of this map is a linear code $C = (X, \mathcal{P}, D)_{L} \subset \mathbb{F}_{q}^{n}$. Using the Riemann-Roch theorem one can estimate the dimension and minimum distance of this code. The authors show that if one wants "good codes" from this construction, that is, linear codes with large dimension and minimum distance relative to their length, one needs curves over finite fields with many rational points.

Some of the highlights of Basic Notions are directly related to the above construction and the study of curves with many rational points. Let $N_{q}(g)$ be the maximum number of points on a curve of genus $g$ defined over $\mathbb{F}_{q}$ and let $A(q) = \limsup_{g \to \infty} \frac{N_{q}(g)}{g}$.  The authors introduce zeta functions of curves over finite fields and discuss the Hasse-Weil bound on $N_{q}(g)$ and some of its improvements. They give a proof of the Drinfeld-Vlǎduţ bound, which states that $A(q) \leq \sqrt{q}-1$. They briefly introduce Garcia-Stichtenoth Towers of function fields and explain how these towers prove that $A(q) = \sqrt{q} − 1$ when $q$ is an even power of a prime.  They show how families of curves with many rational points lead to the existence of good codes, proving the Tsfasman-Vlǎlduţ-Zink bound.

So if that’s all in Basic Notions, what’s in Advanced Chapters? The first major topic tackled by the authors is the study of curves with many rational points. In Basic Notions the authors give hints about where these curves come from, but in the first big section of Advanced Chapters (Chapters 5–7), we get a much fuller picture of this subject. In the introduction to Chapter 5 they list methods to construct such curves:

1. Modular curves of various types;
2. Class field theory constructions;
3. Explicit recursive curve towers;
4. Curves coming from Artin-Schreier or Kummer covers;
5. Other constructions.
The authors discuss each of these approaches, pursuing multiple directions and perspectives at the same time. For example, they explain more geometric aspects of curves along with results about algebraic function fields. The material presented here nicely complements references like Algebraic Function Fields and Codes by Stichtenoth and Algebraic Curves over a Finite Field by Hirschfeld, Korchmáros, and Torres.

In the Preface, the authors say that they are especially interested in connections between different parts of mathematics that come up in this area. It shows! I will give a few examples where the authors highlight connections to other topics. Chapter 10 is about sphere packings and the analogy between good linear codes and dense lattice packings. It gives a different perspective on this topic than Ebeling's nice book Lattices and Codes. Chapter 9 focuses on the decoding of algebraic geometry codes, a topic of much interest at the boundary of pure and applied mathematics. Finally, Chapter 12 gives applications to several areas, including cryptography, algorithms for matrix multiplication, and others.