MinT - Best Known (1, 1+27, s)-Nets in Base 27

Best Known (1, 1+27, s)-Nets in Base 27

(1, 1+27, 38)-Net over F₂₇ — Constructive and digital

Digital (1, 28, 38)-net over F₂₇, using

net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
  - a function field by SÃ©mirat [i]

(1, 1+27, 80)-Net over F₂₇ — Upper bound on s (digital)

There is no digital (1, 28, 81)-net over F₂₇, because

extracting embedded orthogonal array [i] would yield linear OA(27²⁸, 81, F₂₇, 27) (dual of [81, 53, 28]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(27⁵³, 81, F₂₇, 52) (dual of [81, 28, 53]-code), but
  - discarding factors / shortening the dual code would yield linear OA(27⁵³, 56, F₂₇, 52) (dual of [56, 3, 53]-code), but
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

(1, 1+27, 135)-Net in Base 27 — Upper bound on s

There is no (1, 28, 136)-net in base 27, because

19 times m-reduction [i] would yield (1, 9, 136)-net in base 27, but
- extracting embedded orthogonal array [i] would yield OA(27⁹, 136, S₂₇, 8), but
  - the linear programming bound shows that MÂ â‰¥ 35766 310676 101581 380904 / 4611 122641 > 27⁹ [i]