Binary Field Arithmetic

Binary field arithmetic is used in coding. The basic rules are very simple. Addition is as follows:

$\begin{matrix} 0 + 0 = 0 \\ 0 + 1 = 1 \\ 1 + 0 = 1 \\ 1 + 1 = 0 \end{matrix}$ $\begin{matrix} 0 + 0 = 0 \\ 0 + 1 = 1 \\ 1 + 0 = 1 \\ 1 + 1 = 0 \end{matrix}$

This also corresponds to the exclusive OR or XOR operator in digital logic. Subtraction results are identical to addition; there is no distinction between the two operations.

Multiplication is as follows:

$\begin{matrix} 0 \cdot 0 = 0 \\ 0 \cdot 1 = 0 \\ 1 \cdot 0 = 0 \\ 1 \cdot 1 = 1 \end{matrix}$ $\begin{matrix} 0 \cdot 0 = 0 \\ 0 \cdot 1 = 0 \\ 1 \cdot 0 = 0 \\ 1 \cdot 1 = 1 \end{matrix}$

Division is not defined or allowed.

Linearity means that if two input codewords are input to a system, X₁ and X₂, the output result is the same as if the sum of two codewords X₁ + X₂ were input instead. A more mathematical way to express this idea is as follows:

$\begin{matrix} Y 1 = f (X 1) \\ Y 2 = f (X 2) \end{matrix}$ $\begin{matrix} Y 1 = f (X 1) \\ Y 2 = f (X 2) \end{matrix}$

For a linear system

$Y 2 + Y 1 = f (X 2 + X 1)$ $Y 2 + Y 1 = f (X 2 + X 1)$

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Appendix E. Binary Field Arithmetic

Create new playlist

Sign In

Sign Up

Table of Contents for
Appendix E. Binary Field Arithmetic