### 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 Z_{m}, the integers modulo m.

Original language | English (US) |
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Pages (from-to) | 136-143 |

Number of pages | 8 |

Journal | Journal of Combinatorial Theory |

Volume | 9 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1970 |

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### Cite this

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory*, vol. 9, no. 2, pp. 136-143. https://doi.org/10.1016/S0021-9800(70)80020-5

}

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.

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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 -