Problem 37 - Discrete Mathematics with Applications

Question

One of the two distinct de Bruij sequences for binary triplets is 01110100 . Find the other de Bruijn sequence.

Step-by-Step Solution

Answer

The other de Bruijn sequence for binary triplets is the complementary sequence, obtained by swapping each bit in the given sequence: \(10001011\).

Step 1: Identify given sequence

We are given one of the de Bruijn sequences for binary triplets: 01110100.

Step 2: Swap each bit in the given sequence

Go through the given sequence and swap each 0 for a 1 and each 1 for a 0. This will give us the complementary sequence. 01110100 ↓↓↓↓↓↓↓↓ 10001011

Step 3: Write the other de Bruijn sequence

The other de Bruijn sequence for binary triplets is: 10001011.

Key Concepts

Binary Triplets

Binary triplets are sequences consisting of three binary digits, which are simply 0s and 1s. Each triplet represents a possible state or combination in a binary system. For example, in a system that employs binary coding, you can have a number like 000, 001, 010, and so on.
In total, there are eight possible binary triplets, as each digit can be either a 0 or a 1. These possible triplets are:

• 000
• 001
• 010
• 011
• 100
• 101
• 110
• 111

Understanding binary triplets is crucial when dealing with de Bruijn sequences, as these sequences are constructed on the basis of all possible combinations of such binary triplets.

Combinatorial Sequences

Combinatorial sequences are sequences formed using combinations of a set of items, according to specific rules or patterns. These sequences play a key role in fields like graph theory and coding.
A de Bruijn sequence is a specific type of combinatorial sequence. It represents a cyclic sequence in which every possible combination of a certain length appears exactly once. For binary sequences of length three, each possible triplet 000 through 111 appears exactly once. This makes the sequence particularly useful in scenarios such as error detection and computer algorithms because they ensure all possible states or combinations are covered.
De Bruijn sequences are defined by their parameters, such as the length of the subsequence (triplet) and the binary (two-symbol) nature. These parameters make them a fascinating study in the world of mathematics.

Sequence Generation

Sequence generation is the process of creating a series or order of numbers or symbols according to a specific pattern or rule. In the case of de Bruijn sequences, the objective is to generate a sequence that contains every possible combination of a specified length exactly once. In our exercise for binary triplets, the sequences are of length three.
The generation of de Bruijn sequences can be achieved through various algorithms that are designed to systematically and efficiently construct these sequences without repetitions.
One simple way to find the complementary de Bruijn sequence is by inverting the bits (or swapping 0s with 1s) of a known sequence, as demonstrated in the solution: just take the given sequence, swap each digit, and you have the complementary sequence. This inversion technique is a straightforward approach when working with binary systems and can also be useful in other combinatorial contexts.