Intro

---

This is definitions of mathematic related page

Summations

---

In Communication systems, the summation of bits or strings is in a Galios Field (GF)

Example

take GF(2) for example

A	B	Operation	Result
0	0	+/-	0
0	1	+/-	1
1	0	+/-	1
1	1	+/-	0
0	0	x	0
0	1	x	0
1	0	x	0
1	1	x	1

As you can see, the +/- in GF(2) is also xor operation and x is and operation. That is $$a \pm b = a \oplus b \\[1em] a \times b = a \& b$$

Hamming Distance

---

Hamming distance is the number of errors in a communication system.

Example

$$H_d(\text{12345}, \text{22345}) = 1 \\[1em] H_d(\text{12345}, \text{12545}) = 1 \\[1em] H_d(\text{12345}, \text{22355}) = 2 \\[1em] H_d(\text{12345}, \text{54321}) = 4 \\[1em]$$

Galois Field

---

NOTE: This is only a brief introduction to galios field, some of the concepts might be wrong, but is more easy for the learnings.

Field is a group of numbers that are well defined their operations. Galois Field is a group of finite elements, which is also often called finite field, and often use $GF$ to present it. We often put $\Bbb Z$ in Galois Field, since $\Bbb R$ have infinite numbers.

$GF(n)$

This is exactly the same of $$f(x) \equiv x\ mod(n)$$ Take $GF(2)$for example:

3 -> f(3) = 3%2 = 1

Therefore 3 in $GF(2)$ is 1. We can see that the basic operators ( which is +- ) remains the same.

Note: the */ operator are quite different, while $3 \times 2 \equiv 1\ mod(6)$ however $1 / 2 = 0.5$ does not equal to anything.

$GF(n^m)$

It is nearly the same as $GF(n)$, but in a polynomial term. That is, $GF(2^m)$ can be written as $$f(X) \equiv X\ mod(\Sigma^{m}_{i = 0} a_ix^i)$$ where $a_i$ is the coefficient in $GF(n)$ with $a_m \ne 0$ and X is the given polynomial, which also needs to be in $GF(n)$. $\Sigma^{m}_{i = 0} a_ix^i$ can be chosen, and we called this chosen polynomial generator.

Take GF(2^3) for example, we chose generator is $x^3 + x^1 + 1$:

$$x^4 + x^3 + x^1 \rightarrow f(X) = x^2 + x + 1$$

Reference, remember that the +- is xor for every x since there is no need for the carries.

Therefore $x^4 + x^3 + x^1$ in $GF(2^3)$ is $x^2 + x + 1$.

Generator of a Galois Field

A Galois Field is equiavlent to a function below $$f(X) \equiv X\ mod(\Sigma^{m}_{i = 0} a_ix^i)$$ where X is a polynomial and $\Sigma^{m}_{i = 0} a_ix^i$ is chosen. Therefore we call this chosen polynomial ($\Sigma^{m}_{i = 0} a_ix^i$) generator.