Error correcting binary codes

<p>In this paper we consider ways of appending k binary check digits to an n binary digit message word resulting in an n + k sequence of digits called a code word. Determining the k check digits is the "encoding problem." In Chapters 1, 2, 3, and 5, we are primarily concerned with linear codes in which the encoder is a linear transformation of then dimensional vector space containing the message vectors into the vector space of dimension n + k, such that certain errors can be located or at least detected. In Chapter 1, we give the necessary and sufficient conditions for which an (n,k)-code can be constructed such that errors of weight ℓ or less can be corrected. Also the conditions which are necessary and sufficient are given for an (n,k)-code to detect errors of weight ℓ+ l. Chapter 2 develops the Hamming Codes which correct all errors of weight 1. The required field theory is given to construct such an (n ,k)-code. As we may desire to detect errors of weight 2, Chapter 3 develops the construction of an (n,k)-code which is 1-correctable and detects errors of weight 2. Then in Chapters 4 and 5, we consider multiple error correction. In Chapter 4 we develop a nonlinear code which can correct errors of weight ℓ be less, but it has a very low information rate. In Chapter 5 we construct the primitive (.n,k)-codes which are a result of Bose, Chaudhuri, and Hocquenghem which have an improved information rate over the code developed in Chapter 4.</p>

Name thesis:degree_name

Master of Science (MS) in Mathematics

Level thesis:degree_level

Thesis: EWU Only

Discipline thesis:degree_discipline

Mathematics

Year

1972

Author dc:creator

• Gengelbach, Richard R.

• Discrete Mathematics and Combinatorics
• Other Applied Mathematics

Statement dc:rights

• Access perpetually restricted to EWU users with an active EWU NetID

Repository record dc:identifier

https://dc.ewu.edu/theses/794

OAI identifier oai:identifier

oai:dc.ewu.edu:theses-1794