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We present a construction, based on Fourier and Finite Radon Transforms, for families of periodic perfect arrays, each of size $$p\\times p$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u00d7<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for many prime values p<\/jats:italic>. Previously delta functions were used as the discrete projections which, when back-projected, build 2D<\/jats:italic> perfect arrays. Here we employ perfect sequences as the discrete projected views. The base family size is $$p+1$$<\/jats:tex-math>\n \n p<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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 $$-p$$<\/jats:tex-math>\n \n -<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$+p$$<\/jats:tex-math>\n \n +<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, with a zero value at just one location. 2) They have the property of \u2018conjugate\u2019 spatial symmetry, where the value at location (i<\/jats:italic>,\u00a0j<\/jats:italic>) is always the negative of the value at location $$(p-i, p-j)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n p<\/mml:mi>\n -<\/mml:mo>\n i<\/mml:mi>\n ,<\/mml:mo>\n p<\/mml:mi>\n -<\/mml:mo>\n j<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 3) Any change in the value assigned to the array\u2019s origin leaves all of its off-peak autocorrelation values unchanged. 4) A family of $$p+1$$<\/jats:tex-math>\n \n p<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$p\\times p$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u00d7<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> arrays can be compressed to size $$(p+1)^2$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n p<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and each family member can be exactly and rapidly unpacked in a single $$p\\times p$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u00d7<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> decompression pass.<\/jats:p>","DOI":"10.1007\/s10472-023-09903-9","type":"journal-article","created":{"date-parts":[[2023,11,2]],"date-time":"2023-11-02T05:07:00Z","timestamp":1698901620000},"page":"169-182","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Families of multi-level Legendre-like arrays"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8714-1484","authenticated-orcid":false,"given":"Timothy","family":"Petersen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin","family":"Cavy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7758-7140","authenticated-orcid":false,"given":"David","family":"Paganin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Imants","family":"Svalbe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,11,2]]},"reference":[{"key":"9903_CR1","doi-asserted-by":"publisher","first-page":"934","DOI":"10.1109\/TIT.1983.1056744","volume":"29","author":"M Golay","year":"1983","unstructured":"Golay, M.: The merit factor of Legendre sequences (corresp.). 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