The image shows a section on Goppa codes, a type of error-correcting code. Let's break down the key points from the text:

1. Definition of Goppa Codes (9.2.4):

• Let $g(z)$ be a monic polynomial of degree $t$ over $\mathbb{F}_{q^m}$.
• The set $L = \{ y_0, y_1, \dots, y_{n-1} \} \subset \mathbb{F}_{q^m}$ consists of distinct elements, and we also require that $g(y_i) \neq 0$ for all $i$.
• The Goppa code $\Gamma(L, g)$ is defined over the alphabet $\mathbb{F}_q$ using codewords $\mathbf{c} = (c_0, c_1, \dots, c_{n-1})$ that satisfy the following condition:

$\sum_{i=0}^{n-1} \frac{c_i}{z - y_i} \equiv 0 \pmod{g(z)}$ This condition specifies that the sum of the rational functions formed by the codeword components $c_i$ must be congruent to 0 modulo $g(z)$.

Key observation: Goppa codes are linear, which is a desirable property in coding theory as it simplifies analysis and decoding.

2. Example (9.2.6):

• The Goppa polynomial is taken as $g(z) = z^{d-1}$, a degree $d-1$ polynomial.
• The set $L = \{ \beta^{-i} \mid 0 \leq i \leq n - 1 \}$, where $\beta$ is a primitive $n$-th root of unity in $\mathbb{F}_{q^m}$, is chosen.
• With these choices, the resulting Goppa code $\Gamma(L, g)$ is a narrow-sense BCH code of designed distance $d$.

3. Parity Check Matrix:

The connection with section 9.1 is discussed, where the goal is to find a suitable parity check matrix for the Goppa code $\Gamma(L, g)$. Using the earlier results (equations 9.2.5 and 9.2.3), the parity check matrix is presented in the following form:

$\left( \frac{1}{g(y_0)}, \frac{g(z) - g(y_0)}{(z - y_0)g(y_0)}, \dots, \frac{g(z) - g(y_{n-1})}{(z - y_{n-1})g(y_{n-1})} \right)$

Each entry here is interpreted as a column vector, which together form the parity check matrix.

4. Inverse Relations:

• The term $h_j = g(y_j)^{-1}$ is introduced, ensuring $h_j \neq 0$, because $g(y_i) \neq 0$ was part of the definition of Goppa codes.
• If $g(z) = \sum_{i=0}^t g_i z^i$, a general polynomial, then similar properties apply, and the difference of polynomials $g(z) - g(x)$ can be expanded as shown.

In summary:

• Goppa codes are defined using a monic polynomial and a set of distinct elements from a finite field.
• They are linear codes and can be related to BCH codes, with a designed minimum distance $d$.
• A parity check matrix can be constructed based on the properties of the Goppa polynomial and the elements in the set $L$.

Would you like further details on any specific part of this explanation?

5 Related Questions:

1. What makes Goppa codes different from other linear error-correcting codes?
2. How is the designed distance $d$ of the BCH code related to the Goppa polynomial?
3. What are the advantages of using the parity check matrix in decoding Goppa codes?
4. How does the choice of the set $L$ impact the properties of the Goppa code?
5. Could you explain how BCH codes are a special case of Goppa codes?

Tip:

Goppa codes are highly regarded for their error-correction capability, especially in cryptographic applications, due to their algebraic structure and ability to handle multiple errors.