MinT - Best Known (21, 21+69, s)-Nets in Base 4

Best Known (21, 21+69, s)-Nets in Base 4

(21, 21+69, 34)-Net over F₄ — Constructive and digital

Digital (21, 90, 34)-net over F₄, using

net from sequence [i] based on digital (21, 33)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
  - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(21, 21+69, 44)-Net over F₄ — Digital

Digital (21, 90, 44)-net over F₄, using

net from sequence [i] based on digital (21, 43)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
  - a curve from the manYPoints database [i]

(21, 21+69, 91)-Net over F₄ — Upper bound on s (digital)

There is no digital (21, 90, 92)-net over F₄, because

5 times m-reduction [i] would yield digital (21, 85, 92)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁸⁵, 92, F₄, 64) (dual of [92, 7, 65]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁸⁴, 88, F₄, 64) (dual of [88, 4, 65]-code), but
      - Griesmer bound [i]
    2. OA(4⁷, 92, S₄, 4), but
      - discarding factors would yield OA(4⁷, 61, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 16654 > 4⁷ [i]

(21, 21+69, 92)-Net in Base 4 — Upper bound on s

There is no (21, 90, 93)-net in base 4, because

7 times m-reduction [i] would yield (21, 83, 93)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁸³, 93, S₄, 62), but
  - the linear programming bound shows that MÂ â‰¥ 412 815986 320748 811220 279730 438614 536337 297851 038207 836160 / 3 482479 > 4⁸³ [i]