On the Comparative Study of Some Mathematical Tools for Specific Sequences Design

Abstract

In modern communication system, cryptography and automatic test patterns, some specific sequences with strictly defined properties are required to meet the application demands. [1,2,3,4…] These requirements are:

• Good pseudorandom (PN) properties (large period length, uniform distrbution…).
• Low periodic correlation property.
• Low aperiodic correlation property.
• Large linear complexity.
• Large cardinality (number of sequences in the set).

Unfortunately, there is not any set of sequence satisfying all these requirements, despite the fact that a lot of efforts have been given for design such sequences. For this purpose, different mathematical tools have been widely used such as: matrix, d-transform and trace function representations. However, to the best of our knowledge, there has been no any comparative study on these mathematicals tools carried out so far!

In this contribution, we try to fill some parts of this gap by considering some typical applications of these tools. The paper is constructed as below:

• In the introduction, the basic concepts and definitions of matrix, d-transform and trace are given (briefly).
• In section II, some typical applications (for demonstration) will be shown. In this regard, we will give some discussions and suggestions for choosing the appropriate mathematical tools for each application.
• Especially, in section III, we will show the relationship (interchanging or equivalencies) between them.

Keywords: specific sequences, mathematical tool, matrix, d-transform and trace.