MinT - Best Known (6, 91, s)-Nets in Base 16

Best Known (6, 91, s)-Nets in Base 16

(6, 91, 65)-Net over F₁₆ — Constructive and digital

Digital (6, 91, 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, 91, 149)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (6, 91, 150)-net over F₁₆, because

1 times m-reduction [i] would yield digital (6, 90, 150)-net over F₁₆, but
- 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
        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]
        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]

(6, 91, 151)-Net in Base 16 — Upper bound on s

There is no (6, 91, 152)-net in base 16, because

extracting embedded orthogonal array [i] would yield OA(16⁹¹, 152, S₁₆, 85), but
- the linear programming bound shows that MÂ â‰¥ 2 278206 633061 117229 098518 919813 179843 089019 560647 409807 403603 381299 543561 427491 281877 305424 620815 339341 203582 514016 289427 818186 181942 733172 772853 252096 / 60571 801747 214090 732136 958267 349615 729255 > 16⁹¹ [i]