MinT - Best Known (3, 90, s)-Nets in Base 32

Best Known (3, 90, s)-Nets in Base 32

(3, 90, 64)-Net over F₃₂ — Constructive and digital

Digital (3, 90, 64)-net over F₃₂, using

net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
  - a function field by SÃ©mirat [i]

(3, 90, 186)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (3, 90, 187)-net over F₃₂, because

2 times m-reduction [i] would yield digital (3, 88, 187)-net over F₃₂, but
- extracting embedded orthogonal array [i] would yield linear OA(32⁸⁸, 187, F₃₂, 85) (dual of [187, 99, 86]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(32⁸⁷, 91, S₃₂, 85), but
      - the linear programming bound shows that MÂ â‰¥ 3218 248804 644806 290772 103657 191613 826913 486259 137011 802066 676331 137713 410455 220883 164990 453062 699224 855954 368127 744971 436177 620882 948096 / 34443 > 32⁸⁷ [i]
    2. linear OA(32⁹⁹, 187, F₃₂, 96) (dual of [187, 88, 97]-code), but
      - discarding factors / shortening the dual code would yield linear OA(32⁹⁹, 132, F₃₂, 96) (dual of [132, 33, 97]-code), but
        residual code [i] would yield OA(32³, 35, S₃₂, 3), but
        1 times truncation [i] would yield OA(32², 34, S₃₂, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1055 > 32² [i]

(3, 90, 298)-Net in Base 32 — Upper bound on s

There is no (3, 90, 299)-net in base 32, because

1 times m-reduction [i] would yield (3, 89, 299)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁸⁹, 299, S₃₂, 86), but
  - the linear programming bound shows that MÂ â‰¥ 2 360435 264521 435753 863984 728525 460417 636601 778557 757701 268173 418031 085326 973077 668398 961432 098396 263745 738727 715106 031793 750019 589380 141259 838614 726353 354752 / 25927 420448 859780 644999 > 32⁸⁹ [i]