MinT - Best Known (2, 2+55, s)-Nets in Base 27

Best Known (2, 2+55, s)-Nets in Base 27

(2, 2+55, 48)-Net over F₂₇ — Constructive and digital

Digital (2, 57, 48)-net over F₂₇, using

net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
  - a function field by GarcÃa and Quoos [i]

(2, 2+55, 83)-Net over F₂₇ — Upper bound on s (digital)

There is no digital (2, 57, 84)-net over F₂₇, because

1 times m-reduction [i] would yield digital (2, 56, 84)-net over F₂₇, but
- extracting embedded orthogonal array [i] would yield linear OA(27⁵⁶, 84, F₂₇, 54) (dual of [84, 28, 55]-code), but
  - residual code [i] would yield OA(27², 29, S₂₇, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 755 > 27² [i]

(2, 2+55, 203)-Net in Base 27 — Upper bound on s

There is no (2, 57, 204)-net in base 27, because

8 times m-reduction [i] would yield (2, 49, 204)-net in base 27, but
- extracting embedded orthogonal array [i] would yield OA(27⁴⁹, 204, S₂₇, 47), but
  - the linear programming bound shows that MÂ â‰¥ 44445 526432 553148 551803 587352 645711 159091 434363 391143 128643 749994 239134 847577 444125 / 3 214315 016317 > 27⁴⁹ [i]