91Ó°ÊÓ

A coding system encodes messages using strings of base 4 digits (that is, digits from the set \(\\{0,1,2,3\\} )\) . A codeword is valid if and only if it contains an even number of 0 \(\mathrm{s}\) an even number of 1 \(\mathrm{s}\) . Let \(a_{n}\) equal the number of valid codewords of length \(n .\) Furthermore, let \(b_{n}, c_{n},\) and \(d_{n}\) equal the number of strings of base 4 digits of length \(n\) with an even number of 0 \(\mathrm{s}\) and an odd number of \(1 \mathrm{s},\) with an odd number of 0 \(\mathrm{s}\) and an even number of \(1 \mathrm{s},\) and with an odd number of 0 \(\mathrm{s}\) and an odd number of 1 \(\mathrm{s}\) , respectively. a) Show that \(d_{n}=4^{n}-a_{n}-b_{n}-c_{n}\) . Use this to show that \(a_{n+1}=2 a_{n}+b_{n}+c_{n}, b_{n+1}=b_{n}-c_{n}+4^{n},\) and \(c_{n+1}=c_{n}-b_{n}+4^{n} .\) b) What are \(a_{1}, b_{1}, c_{1},\) and \(d_{1} ?\) c) Use parts (a) and (b) to find \(a_{3}, b_{3}, c_{3},\) and \(d_{3}\) d) Use the recurrence relations in part (a), together with the initial conditions in part (b), to set up three equa- tions relating the generating functions \(A(x), B(x),\) and \(C(x)\) for the sequences \(\left\\{a_{n}\right\\},\left\\{b_{n}\right\\},\) and \(\left\\{c_{n}\right\\},\) respec- tively. e) Solve the system of equations from part (d) to get ex- plicit formulae for \(A(x), B(x),\) and \(C(x)\) and use these to get explicit formulae for \(a_{n}, b_{n}, c_{n},\) and \(d_{n}\)

Step by step solution

01

- Understand and Define the Problem

Given the definitions of valid codewords and the conditions for validity based on the counts of 0s and 1s, first understand the basic criteria for each sequence. We need to prove some properties and relationships among these sequences.

02

- Prove Equation for d_n

We start by proving that the total number of strings of length n in base 4 is given by \(4^n\). Since we have all possibilities included in a_n, b_n, c_n, and d_n, it must be true that: \[d_n = 4^n - a_n - b_n - c_n\]

03

- Establish Recurrence Relations

Using the definitions and criteria for valid strings: a) A valid string of length \(n+1\) can be formed by adding one digit to valid strings of length \(n\). Therefore, \( a_{n+1} = 2a_n + b_n + c_n \)b) For sequences with an odd number of ones and even number of zeros, and similar characteristics: \( b_{n+1} = b_n - c_n + 4^n \)c) A similar logic implies: \( c_{n+1} = c_n - b_n + 4^n \)

04

- Find Initial Values

For \(n = 1\): Calculate the values * \(a_1\): Valid strings are 0 or 2, perfectly balanced and valid by our criteria. * \(b_1\): Valid strings are 1 and 3, adjusting for counting rules.* \(c_1\): matches certain criteria with odd/even combinations* \(d_1\): is all other resulting combinations not covered in the above specific scenarios

05

- Calculate Specific Values

Use the formulas derived and initial values to calculate updated values for sequences up to n = 3. Use recursion: \(a_2, b_2, c_2, d_2\);\(a_3, b_3, c_3, d_3\) follow similar logic steps iteratively.

06

- Setup Generating Functions

Define Functions: \( A(x) = \sum a_n x^n \);\( B(x) = \sum b_n x^n \);\( C(x) = \sum c_n x^n \). Use given recurrences to build function relationships.

07

- Solve Generating Function System

Solve the generated equations from Step 6; Use algebraic manipulations to extract explicit forms of \(A(x), B(x), C(x)\).Extract function terms defining new sequences.

08

- Derive Explicit Formulas

Obtain terms \(a_n, b_n, c_n\) using generating functions. Verify these satisfy all initial recursions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

recurrence relations

Recurrence relations are equations that define sequences based on previous terms. In this problem, we use recurrence relations to find the number of valid codewords of certain lengths by breaking down the problem into smaller components.
Consider the given sequences:
\[a_{n}\] is the number of valid codewords of length \(n\) with the required even number of 0s and 1s.
\[b_{n}\] is the number of codewords with an even number of 0s and an odd number of 1s.
\[c_{n}\] is the number of codewords with an odd number of 0s and an even number of 1s.
\[d_{n}\] is the number of codewords with an odd number of both 0s and 1s.

From these definitions, we derive that the total number of codewords of length \(n\) is given by:
\[d_{n} = 4^n - a_{n} - b_{n} - c_{n}\]

Next, we use recurrence relations to express \(a_{n+1}\), \(b_{n+1}\), and \(c_{n+1}\) in terms of \(a_{n}\), \(b_{n}\), and \(c_{n}\). These are derived using combinatorial arguments:
\[a_{n+1} = 2a_{n} + b_{n} + c_{n}\]
\[b_{n+1} = b_{n} - c_{n} + 4^n\]
\[c_{n+1} = c_{n} - b_{n} + 4^n\]

We also need initial values to start solving these recurrence relations.

base 4 digits

Base 4 digits refer to a numeral system that uses four symbols: 0, 1, 2, and 3. Unlike the common base 10 system which uses digits from 0 to 9, base 4 only has these four digits.

In this problem, base 4 digits are used to encode messages into codewords. Each position in the codeword can be any of these four digits.

When working with base 4, each digit represents a power of 4 based on its position. For a codeword of length \(n\), there are \(4^n\) possible combinations.

We need to consider the specific counts of each digit to determine whether a codeword is valid according to our rules. For instance, a valid codeword must contain an even number of 0s and 1s. Using base 4 digits allows us to utilize these constraints effectively in our counting and analysis.

combinatorial enumeration

Combinatorial enumeration is a technique used to count or enumerate objects that meet certain criteria. In this exercise, we use this method to find the number of valid codewords based on the given rules.

To systematically count the valid codewords, we group them based on their properties, such as having an even or odd number of 0s and 1s. These groups help us create recurrence relations and identify the counts of each specific type of codeword.

Here’s how the combinations are grouped:
1. \(a_{n}\): Codewords with even 0s and even 1s
2. \(b_{n}\): Codewords with even 0s and odd 1s
3. \(c_{n}\): Codewords with odd 0s and even 1s
4. \(d_{n}\): Codewords with odd 0s and odd 1s

By breaking down the problem into different cases, we can solve each part independently. Then, we can combine the solutions to count the total number of valid codewords.

valid codewords

A valid codeword in this context meets specific criteria, having even numbers of certain digits. For instance:

1. A codeword of length \(n\) is valid if it contains an even number of 0s and an even number of 1s. These codewords are counted in the sequence \(a_{n}\).
2. Codewords with an even number of 0s and an odd number of 1s belong to the sequence \(b_{n}\).
3. Codewords with an odd number of 0s and an even number of 1s fall into sequence \(c_{n}\).
4. Finally, codewords with an odd number of both 0s and 1s are in sequence \(d_{n}\).

To determine these counts, we analyze the properties of the codewords and use recurrence relations derived from these properties. This systematic approach helps in categorizing and reliably counting valid codewords of any length \(n\).

Understanding the criteria for validity is crucial as it forms the foundation of setting up our recurrence relations and in turn, finding the exact number of valid codewords for any code length.