Integers and Modern Applications for the Uninitiated

A code is a system of rules for changing information from one form to another. When we change given information into another representation, we are encoding. When we change it back to the original representation, we are decoding. We describe the rules for encoding and decoding with functions. The encoding function must be invertible so decoding can recover the original information.

In this section we convert text into a sequence of numbers. Commonly used character encodings are ASCII (American Standard Code for Information Interchange) and Unicode. ASCII uses 128 printable and control characters and was standardized in 1963 by ASA (American Standards Association). Unicode can handle the characters in most of the world’s writing systems.

To convert text into a sequence of numbers, we use the encoding function \(C\) and its inverse, the decoding function \(C^{-1}\text{,}\) given in Figure 8.1.1. In practice, we need only one table. Instead of writing a separate table for \(C^{-1}\text{,}\) we can read the table for \(C\) from right to left.

To encode words or longer parts of text, we apply the encoding function character by character. This gives a sequence of numbers, or more exactly, a sequence of elements of \(\Z_{27}\text{.}\) We separate these numbers with commas.

The video in Figure 8.1.2 explains in detail how to use the encoding function \(C\) and its inverse, the decoding function \(C^{-1}\text{.}\)

In the following we give examples on how to apply the encoding and decoding functions.

To recover text from a sequence of numbers, or more exactly, from a sequence of elements of \(\Z_{27}\text{,}\) we apply the decoding function \(C^{-1}\) to each number.

In the following problems, we no longer explicitly evaluate the encoding function \(C\) or its inverse, the decoding function \(C^{-1}\text{.}\) The encoding and decoding process is a simple lookup in Figure 8.1.1.