MinT - Best Known (6, 6+82, s)-Nets in Base 16

Best Known (6, 6+82, s)-Nets in Base 16

(6, 6+82, 65)-Net over F₁₆ — Constructive and digital

Digital (6, 88, 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]

(6, 6+82, 174)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (6, 88, 175)-net over F₁₆, because

2 times m-reduction [i] would yield digital (6, 86, 175)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16⁸⁶, 175, F₁₆, 80) (dual of [175, 89, 81]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(16⁸⁵, 92, F₁₆, 80) (dual of [92, 7, 81]-code), but
      - constructionÂ Y1 [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]
        OA(16⁷, 92, S₁₆, 6), but
        discarding factors would yield OA(16⁷, 80, S₁₆, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 278 002201 > 16⁷ [i]
    2. OA(16⁸⁹, 175, S₁₆, 83), but
      - discarding factors would yield OA(16⁸⁹, 165, S₁₆, 83), but
        the linear programming bound shows that MÂ â‰¥ 15352 684767 685611 086882 809761 984160 188561 571509 004484 738285 092713 831920 008077 971129 525920 055364 921813 182353 415776 911815 913281 106245 860502 560348 270257 729557 233664 / 103836 312412 686023 017550 716368 186316 554556 371946 618739 > 16⁸⁹ [i]

(6, 6+82, 176)-Net in Base 16 — Upper bound on s

There is no (6, 88, 177)-net in base 16, because

1 times m-reduction [i] would yield (6, 87, 177)-net in base 16, but
- extracting embedded orthogonal array [i] would yield OA(16⁸⁷, 177, S₁₆, 81), but
  - the linear programming bound shows that MÂ â‰¥ 1979 490265 767149 060480 289641 042414 409901 982668 709551 972013 109722 618112 596178 616945 021767 047036 693222 515191 448031 188325 734085 173872 061467 835572 927146 914505 293824 / 3 430222 078227 205946 190279 817987 259978 700370 182828 419579 > 16⁸⁷ [i]