Question

A code for \(a, b, c, d, e\) is given by \(a: 00, b: 01\) c: \(101, d: x 10,\) e: \(y z 1,\) where \(x, y, z\) are in 0,1 Determine \(x, y\) and \(z\) so that the given code is a prefix code.

Short answer

Hence, the values of \(x\), \(y\), and \(z\) that will ensure the given code is a prefix code are \(x=0\), \(y=1\), and \(z=1\) respectively.

Step-by-step solution

Understanding the constraints

Given the codes for \(a\), \(b\), and \(c\), we gauge the values 'x', 'y', and 'z' cannot take. From these codes, now, 'x' cannot be '1' because '10' would then be a prefix for '101', and '1' by itself is already a prefix for '101'. So, \(x = 0\).

Finding the possible values for \(y\)

Now let's find the possible values for \(y\). Here, we know that \(y\) cannot be '0' because '0' would make \(y 0\) a prefix for '00' and '01'. Therefore, \(y = 1\).

Finding the possible values for \(z\)

Lastly, we determine \(z\). Now, \(z\) cannot be '0' because '10' is already a code. So, \(z = 1\) here.

Key concepts

Coding Theory

Coding theory is a critical branch within the field of communication and computer science that revolves around the creation of efficient and reliable methods for transmitting data. Its objective is to find ways in which information can be represented with the use of codes so that it can be transmitted or stored with the requirement of using as little space as possible, while also being safeguarded against errors.

Within this context, prefix codes play a significant role. They are a type of code system where no code word is a prefix of another. This is crucial because it ensures that there is no ambiguity in decoding the received sequence of bits. Prefix codes are also referred to as instantaneous codes, due to their decodable nature without the need for a delimiting symbol or lookahead. Huffman coding is an exemplary algorithm in coding theory that constructs such prefix codes, offering efficient compression for various applications. Understanding how to construct a proper prefix code, as in our exercise, is a fundamental exercise in coding theory.

Algorithms

Algorithms are step-by-step procedures or formulas for solving problems. They are essential in computer science, enabling programmers to devise clear instructions, which a computer can follow to perform a specific task. Algorithms come in various shapes and sizes, tailored to solve particular problems or to carry out specific operations within a wide range of applications.

In the context of our exercise related to coding theory, the algorithmic challenge involves determining values of 'x', 'y', and 'z' that transform an undefined code into a valid prefix code. This involves a process of reasoning through the constraints that define prefix codes, and excluding possibilities that would infringe on those constraints. As demonstrated in the step-by-step solution, each step uses logic to eliminate impossible code combinations systematically, thus narrowing down to the valid code values. Algorithms like these are the bedrock of problem-solving in computer science.

Binary Codes

Binary codes are the most basic form of data representation in computing, using only two symbols, typically 0 and 1, called bits. This binary system is the foundation of virtually all modern computer systems and communication protocols because it aligns with the two-state principle of electronic devices, where a state can be off (0) or on (1).

In reference to the exercise we're exploring, codes for characters 'a' through 'e' are represented as sequences of binary digits. A code is uniquely identifiable when sequenced properly, just as binary representation allows for clear and unambiguous communication between computing systems. When 'x', 'y', and 'z' are determined such that no complete code is the prefix of any other, the binary codes ensure that each symbol can be unambiguously decoded, which is the primary objective of a well-designed binary coding system.