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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Gengelbach, Richard R."],"institution":null,"degree_name":"Master of Science (MS) in Mathematics","degree_level":"Thesis: EWU Only","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1972,"date_issued":"1972-01-01T08:00:00Z","date_published":"1972-01-01T08:00:00Z","updated_at":"2026-07-24T02:13:07Z","subjects":["Discrete Mathematics and Combinatorics","Other Applied Mathematics"],"languages":[],"rights":["Access perpetually restricted to EWU users with an active EWU NetID"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.ewu.edu/theses/794","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Gengelbach, Richard R."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis: EWU Only"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS) in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Discrete Mathematics and Combinatorics","Other Applied Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Access perpetually restricted to EWU users with an active EWU NetID"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.ewu.edu/theses/794"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Error correcting binary codes"]}]}],"canonical_facts":{"dc:creator":["Gengelbach, Richard R."],"dc:description.abstract":["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.
"],"dc:identifier":["https://dc.ewu.edu/theses/794"],"dc:rights":["Access perpetually restricted to EWU users with an active EWU NetID"],"dc:subject":["Discrete Mathematics and Combinatorics","Other Applied Mathematics"],"dc:title":["Error correcting binary codes"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis: EWU Only"],"thesis:degree_name":["Master of Science (MS) in Mathematics"]},"updated_at":"2026-07-24T02:13:07Z"}