Orthogonal sets of vectors over Zm

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

The theory of Hadamard matrices is concerned with finding maximal sets of orthogonal vectors whose components are +1's and -1's. A natural generalization of this problem is to allow entries other than ±1's or to allow entries from some fixed field or ring. In the case of an arbitrary field or ring there may be "orthogonal" sets of "vectors" whose cardinality exceeds the "dimension" of the space. The purpose of this paper is to obtain an explicit formula for the maximum cardinality of such an orthogonal set when the ring involved is Zm, the integers modulo m.

Original languageEnglish (US)
Pages (from-to)136-143
Number of pages8
JournalJournal of Combinatorial Theory
Volume9
Issue number2
DOIs
StatePublished - Jan 1 1970

Fingerprint

Set of vectors
Ring
Cardinality
Hadamard Matrix
Modulo
Explicit Formula
Exceed
Integer
Arbitrary

Cite this

Orthogonal sets of vectors over Zm . / Zame, Alan.

In: Journal of Combinatorial Theory, Vol. 9, No. 2, 01.01.1970, p. 136-143.

Research output: Contribution to journalArticle

@article{5fa213223f6e4503b400770878888d3f,
title = "Orthogonal sets of vectors over Zm",
abstract = "The theory of Hadamard matrices is concerned with finding maximal sets of orthogonal vectors whose components are +1's and -1's. A natural generalization of this problem is to allow entries other than ±1's or to allow entries from some fixed field or ring. In the case of an arbitrary field or ring there may be {"}orthogonal{"} sets of {"}vectors{"} whose cardinality exceeds the {"}dimension{"} of the space. The purpose of this paper is to obtain an explicit formula for the maximum cardinality of such an orthogonal set when the ring involved is Zm, the integers modulo m.",
author = "Alan Zame",
year = "1970",
month = "1",
day = "1",
doi = "10.1016/S0021-9800(70)80020-5",
language = "English (US)",
volume = "9",
pages = "136--143",
journal = "Journal of Combinatorial Theory",
issn = "0021-9800",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - Orthogonal sets of vectors over Zm

AU - Zame, Alan

PY - 1970/1/1

Y1 - 1970/1/1

N2 - The theory of Hadamard matrices is concerned with finding maximal sets of orthogonal vectors whose components are +1's and -1's. A natural generalization of this problem is to allow entries other than ±1's or to allow entries from some fixed field or ring. In the case of an arbitrary field or ring there may be "orthogonal" sets of "vectors" whose cardinality exceeds the "dimension" of the space. The purpose of this paper is to obtain an explicit formula for the maximum cardinality of such an orthogonal set when the ring involved is Zm, the integers modulo m.

AB - The theory of Hadamard matrices is concerned with finding maximal sets of orthogonal vectors whose components are +1's and -1's. A natural generalization of this problem is to allow entries other than ±1's or to allow entries from some fixed field or ring. In the case of an arbitrary field or ring there may be "orthogonal" sets of "vectors" whose cardinality exceeds the "dimension" of the space. The purpose of this paper is to obtain an explicit formula for the maximum cardinality of such an orthogonal set when the ring involved is Zm, the integers modulo m.

UR - http://www.scopus.com/inward/record.url?scp=58149412622&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58149412622&partnerID=8YFLogxK

U2 - 10.1016/S0021-9800(70)80020-5

DO - 10.1016/S0021-9800(70)80020-5

M3 - Article

AN - SCOPUS:58149412622

VL - 9

SP - 136

EP - 143

JO - Journal of Combinatorial Theory

JF - Journal of Combinatorial Theory

SN - 0021-9800

IS - 2

ER -