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

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

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

Digital (2, 55, 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, 55, 109)-Net over F₂₇ — Upper bound on s (digital)

There is no digital (2, 55, 110)-net over F₂₇, because

1 times m-reduction [i] would yield digital (2, 54, 110)-net over F₂₇, but
- extracting embedded orthogonal array [i] would yield linear OA(27⁵⁴, 110, F₂₇, 52) (dual of [110, 56, 53]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(27⁵³, 56, F₂₇, 52) (dual of [56, 3, 53]-code), but
      - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. linear OA(27⁵⁶, 110, F₂₇, 54) (dual of [110, 54, 55]-code), but
      - discarding factors / shortening the dual code 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, 55, 203)-Net in Base 27 — Upper bound on s

There is no (2, 55, 204)-net in base 27, because

6 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]