Families of multi-level Legendre-like arrays Article Swipe
Timothy C. Petersen
,
Benjamin Cavy
,
David M. Paganin
,
Imants Svalbe
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1007/s10472-023-09903-9
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1007/s10472-023-09903-9
Families of new, multi-level integer 2 D arrays are introduced here as an extension of the well-known binary Legendre sequences that are derived from quadratic residues. We present a construction, based on Fourier and Finite Radon Transforms, for families of periodic perfect arrays, each of size $$p\times p$$ for many prime values p . Previously delta functions were used as the discrete projections which, when back-projected, build 2 D perfect arrays. Here we employ perfect sequences as the discrete projected views. The base family size is $$p+1$$ . All members of these multi-level array families have perfect autocorrelation and constant, minimal cross-correlation. Proofs are given for four useful and general properties of these new arrays. 1) They are comprised of odd integers, with values between at most $$-p$$ and $$+p$$ , with a zero value at just one location. 2) They have the property of ‘conjugate’ spatial symmetry, where the value at location ( i , j ) is always the negative of the value at location $$(p-i, p-j)$$ . 3) Any change in the value assigned to the array’s origin leaves all of its off-peak autocorrelation values unchanged. 4) A family of $$p+1$$ , $$p\times p$$ arrays can be compressed to size $$(p+1)^2$$ and each family member can be exactly and rapidly unpacked in a single $$p\times p$$ decompression pass.
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- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/s10472-023-09903-9
- https://link.springer.com/content/pdf/10.1007/s10472-023-09903-9.pdf
- OA Status
- hybrid
- Cited By
- 1
- References
- 34
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- 10
- OpenAlex ID
- https://openalex.org/W4388228996
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https://openalex.org/W4388228996Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1007/s10472-023-09903-9Digital Object Identifier
- Title
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Families of multi-level Legendre-like arraysWork title
- Type
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2023Year of publication
- Publication date
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2023-11-02Full publication date if available
- Authors
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Timothy C. Petersen, Benjamin Cavy, David M. Paganin, Imants SvalbeList of authors in order
- Landing page
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https://doi.org/10.1007/s10472-023-09903-9Publisher landing page
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https://link.springer.com/content/pdf/10.1007/s10472-023-09903-9.pdfDirect link to full text PDF
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YesWhether a free full text is available
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hybridOpen access status per OpenAlex
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https://link.springer.com/content/pdf/10.1007/s10472-023-09903-9.pdfDirect OA link when available
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Algorithm, Mathematics, Computer scienceTop concepts (fields/topics) attached by OpenAlex
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1Total citation count in OpenAlex
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2025: 1Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.Legendre | 19 |
| abstract_inverted_index.assigned | 221 |
| abstract_inverted_index.discrete | 70, 87 |
| abstract_inverted_index.families | 39, 111 |
| abstract_inverted_index.location | 183, 197 |
| abstract_inverted_index.negative | 192 |
| abstract_inverted_index.off-peak | 230 |
| abstract_inverted_index.periodic | 41 |
| abstract_inverted_index.property | 174 |
| abstract_inverted_index.unpacked | 287 |
| abstract_inverted_index.$$p\times | 47, 248, 291 |
| abstract_inverted_index.<mml:math | 49, 96, 145, 154, 200, 239, 250, 265, 293 |
| abstract_inverted_index.array’s | 224 |
| abstract_inverted_index.comprised | 135 |
| abstract_inverted_index.constant, | 116 |
| abstract_inverted_index.extension | 14 |
| abstract_inverted_index.functions | 65 |
| abstract_inverted_index.integers, | 138 |
| abstract_inverted_index.location. | 169 |
| abstract_inverted_index.projected | 88 |
| abstract_inverted_index.quadratic | 25 |
| abstract_inverted_index.residues. | 26 |
| abstract_inverted_index.sequences | 20, 84 |
| abstract_inverted_index.symmetry, | 178 |
| abstract_inverted_index.<mml:mrow> | 51, 98, 147, 156, 202, 241, 252, 268, 295 |
| abstract_inverted_index.<mml:msup> | 267 |
| abstract_inverted_index.Previously | 63 |
| abstract_inverted_index.compressed | 261 |
| abstract_inverted_index.introduced | 10 |
| abstract_inverted_index.properties | 127 |
| abstract_inverted_index.unchanged. | 233 |
| abstract_inverted_index.well-known | 17 |
| abstract_inverted_index.$$(p+1)^2$$ | 264 |
| abstract_inverted_index.</mml:math> | 56, 103, 151, 160, 213, 246, 257, 277, 300 |
| abstract_inverted_index.</mml:mrow> | 55, 102, 150, 159, 212, 245, 256, 274, 299 |
| abstract_inverted_index.</mml:msup> | 276 |
| abstract_inverted_index.Transforms, | 37 |
| abstract_inverted_index.multi-level | 4, 109 |
| abstract_inverted_index.projections | 71 |
| abstract_inverted_index.construction, | 30 |
| abstract_inverted_index.decompression | 301 |
| abstract_inverted_index.autocorrelation | 114, 231 |
| abstract_inverted_index.back-projected, | 74 |
| abstract_inverted_index.‘conjugate’ | 176 |
| abstract_inverted_index.<mml:mi>i</mml:mi> | 206 |
| abstract_inverted_index.<mml:mi>j</mml:mi> | 210 |
| abstract_inverted_index.<mml:mi>p</mml:mi> | 52, 54, 99, 149, 158, 204, 208, 242, 253, 255, 270, 296, 298 |
| abstract_inverted_index.<mml:mn>1</mml:mn> | 101, 244, 272 |
| abstract_inverted_index.<mml:mn>2</mml:mn> | 275 |
| abstract_inverted_index.<mml:mo>(</mml:mo> | 203, 269 |
| abstract_inverted_index.<mml:mo>)</mml:mo> | 211, 273 |
| abstract_inverted_index.<mml:mo>+</mml:mo> | 100, 157, 243, 271 |
| abstract_inverted_index.<mml:mo>,</mml:mo> | 207 |
| abstract_inverted_index.<mml:mo>-</mml:mo> | 148, 205, 209 |
| abstract_inverted_index.cross-correlation. | 118 |
| abstract_inverted_index.<mml:mo>×</mml:mo> | 53, 254, 297 |
| abstract_inverted_index.xmlns:mml="http://www.w3.org/1998/Math/MathML"> | 50, 97, 146, 155, 201, 240, 251, 266, 294 |
| cited_by_percentile_year.max | 95 |
| cited_by_percentile_year.min | 91 |
| corresponding_author_ids | https://openalex.org/A5071007453 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 4 |
| corresponding_institution_ids | https://openalex.org/I56590836 |
| citation_normalized_percentile.value | 0.5966189 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |