MinT - Best Known (1, 1+67, s)-Nets in Base 27

Best Known (1, 1+67, s)-Nets in Base 27

(1, 1+67, 38)-Net over F₂₇ — Constructive and digital

Digital (1, 68, 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+67, 55)-Net over F₂₇ — Upper bound on s (digital)

There is no digital (1, 68, 56)-net over F₂₇, because

15 times m-reduction [i] would yield digital (1, 53, 56)-net over F₂₇, but
- extracting embedded orthogonal array [i] 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+67, 56)-Net in Base 27 — Upper bound on s

There is no (1, 68, 57)-net in base 27, because

13 times m-reduction [i] would yield (1, 55, 57)-net in base 27, but
- extracting embedded orthogonal array [i] would yield OA(27⁵⁵, 57, S₂₇, 54), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 430 023359 390034 222082 732011 948356 798311 147247 214997 695270 038813 781532 497547 421283 / 55 > 27⁵⁵ [i]