MinT - Best Known (90−84, 90, s)-Nets in Base 16

Best Known (90−84, 90, s)-Nets in Base 16

(90−84, 90, 65)-Net over F₁₆ — Constructive and digital

Digital (6, 90, 65)-net over F₁₆, using

net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
  - the Hermitian function field over F₁₆ [i]

(90−84, 90, 149)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (6, 90, 150)-net over F₁₆, because

extracting embedded orthogonal array [i] would yield linear OA(16⁹⁰, 150, F₁₆, 84) (dual of [150, 60, 85]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(16⁸⁹, 94, F₁₆, 84) (dual of [94, 5, 85]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(16⁸⁸, 90, S₁₆, 84), but
        4 times truncation [i] would yield OA(16⁸⁴, 86, S₁₆, 80), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 4 479489 484355 608421 114884 561136 888556 243290 994469 299069 799978 201927 583742 360321 890761 754986 543214 231552 / 27 > 16⁸⁴ [i]
      2. OA(16⁵, 94, S₁₆, 4), but
        the linear programming bound shows that MÂ â‰¥ 2179 760128 / 2071 > 16⁵ [i]
  2. OA(16⁶⁰, 150, S₁₆, 56), but
    - discarding factors would yield OA(16⁶⁰, 147, S₁₆, 56), but
      - the linear programming bound shows that MÂ â‰¥ 939765 966922 797477 814596 181274 291363 410391 878783 132568 319700 182987 088048 908156 194099 962624 995855 466829 447168 / 521248 355323 503233 415469 012835 684063 > 16⁶⁰ [i]

(90−84, 90, 155)-Net in Base 16 — Upper bound on s

There is no (6, 90, 156)-net in base 16, because

extracting embedded orthogonal array [i] would yield OA(16⁹⁰, 156, S₁₆, 84), but
- the linear programming bound shows that MÂ â‰¥ 1257 606533 014336 795226 389703 729813 696829 495404 797141 393550 178599 015967 879273 142761 715793 695948 813145 352197 147287 458391 303937 650589 708999 508795 326464 / 525 287267 173286 237173 455069 701316 546399 > 16⁹⁰ [i]