{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71245"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71245","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Topics in Coding Theory: 1. The A(,s)(n,d) Problem in The Plotkin Region. 2. Number of Information Symbols in a Bch Code","abstract":"In chapter 1 we investigate the A(,s)(n,d) problem in the Plotkin Region. The problem is to finding the maximum number of codewords in a code on an alphabet with s symbols that has length n and minimum Hamming distance d. The Plotkin Region is the set of n and d such that sd > t(s - 1)n. We define Generalized Hadamard matrices, using a notion of orthogonality over a group, and show that these matrices give rise to certain A(,s)(n,d) codes. Two general constructions for A(,s)(n,d) codes, which apparently do not depend on Hadamard matrices, are given. Lastly we discuss a computer implemented algorithm to search for A(,3)(15,11).","abstract_html":"In chapter 1 we investigate the A(,s)(n,d) problem in the Plotkin Region. The problem is to finding the maximum number of codewords in a code on an alphabet with s symbols that has length n and minimum Hamming distance d. The Plotkin Region is the set of n and d such that sd > t(s - 1)n. We define Generalized Hadamard matrices, using a notion of orthogonality over a group, and show that these matrices give rise to certain A(,s)(n,d) codes. Two general constructions for A(,s)(n,d) codes, which apparently do not depend on Hadamard matrices, are given. Lastly we discuss a computer implemented algorithm to search for A(,3)(15,11).","abstract_has_math":false,"creators":["Merkey, Phillip Roy"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:16Z","date_published":"2014-12-16T06:18:16Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8701565"],"render_values":[{"text":"(UMI)AAI8701565","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71245","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Merkey, Phillip Roy"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:16Z","10000-01-01","1986"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71245","(UMI)AAI8701565"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In chapter 1 we investigate the A(,s)(n,d) problem in the Plotkin Region. The problem is to finding the maximum number of codewords in a code on an alphabet with s symbols that has length n and minimum Hamming distance d. The Plotkin Region is the set of n and d such that sd > t(s - 1)n. We define Generalized Hadamard matrices, using a notion of orthogonality over a group, and show that these matrices give rise to certain A(,s)(n,d) codes. Two general constructions for A(,s)(n,d) codes, which apparently do not depend on Hadamard matrices, are given. Lastly we discuss a computer implemented algorithm to search for A(,3)(15,11).","In chapter 2 we discuss the problem of finding the number of information symbols in a BCH code with design parameters n, b and d. This work generalizes the work of Berlekamp where he completely answers the problem for the case of the "simple" BCH codes (the case b = 0). In the general case (where b is arbitrary) we show the problem to be equivalent to counting certain walks in a directed graph. By considering the adjacency matrix for the graph we find a efficient method of computing I(n,d,b) and we find the asymptotic growth of I(n,d,b) as n, d and b increase while the ratios d/n and b/n are fixed.","Made available in DSpace on 2014-12-16T06:18:16Z (GMT). No. of bitstreams: 1 8701565.pdf: 5094327 bytes, checksum: 62675a94d0e7d276ce4988d09a4465df (MD5) Previous issue date: 1986","Embargo set by: Seth Robbins for item 71411 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","147 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1986."]},{"key":"dc:title","label":"Title","values":["Topics in Coding Theory: 1. The A(,s)(n,d) Problem in The Plotkin Region. 2. Number of Information Symbols in a Bch Code"]}]}],"canonical_facts":{"dc:creator":["Merkey, Phillip Roy"],"dc:date":["2014-12-16T06:18:16Z","10000-01-01","1986"],"dc:description":["In chapter 1 we investigate the A(,s)(n,d) problem in the Plotkin Region. The problem is to finding the maximum number of codewords in a code on an alphabet with s symbols that has length n and minimum Hamming distance d. The Plotkin Region is the set of n and d such that sd > t(s - 1)n. We define Generalized Hadamard matrices, using a notion of orthogonality over a group, and show that these matrices give rise to certain A(,s)(n,d) codes. Two general constructions for A(,s)(n,d) codes, which apparently do not depend on Hadamard matrices, are given. Lastly we discuss a computer implemented algorithm to search for A(,3)(15,11).","In chapter 2 we discuss the problem of finding the number of information symbols in a BCH code with design parameters n, b and d. This work generalizes the work of Berlekamp where he completely answers the problem for the case of the "simple" BCH codes (the case b = 0). In the general case (where b is arbitrary) we show the problem to be equivalent to counting certain walks in a directed graph. By considering the adjacency matrix for the graph we find a efficient method of computing I(n,d,b) and we find the asymptotic growth of I(n,d,b) as n, d and b increase while the ratios d/n and b/n are fixed.","Made available in DSpace on 2014-12-16T06:18:16Z (GMT). No. of bitstreams: 1 8701565.pdf: 5094327 bytes, checksum: 62675a94d0e7d276ce4988d09a4465df (MD5) Previous issue date: 1986","Embargo set by: Seth Robbins for item 71411 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","147 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1986."],"dc:identifier":["http://hdl.handle.net/2142/71245","(UMI)AAI8701565"],"dc:subject":["Mathematics"],"dc:title":["Topics in Coding Theory: 1. The A(,s)(n,d) Problem in The Plotkin Region. 2. Number of Information Symbols in a Bch Code"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}