MinT - Best Known (33, 33+77, s)-Nets in Base 27

Best Known (33, 33+77, s)-Nets in Base 27

(33, 33+77, 114)-Net over F₂₇ — Constructive and digital

Digital (33, 110, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 110, 114)-net over F₂₇, using
- net from sequence [i] based on digital (23, 113)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 23 and N(F) ≥ 114, using
    - an explicitly constructive algebraic function field [i]

(33, 33+77, 150)-Net in Base 27 — Constructive

(33, 110, 150)-net in base 27, using

27² times duplication [i] based on (31, 108, 150)-net in base 27, using
- base change [i] based on digital (4, 81, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(33, 33+77, 220)-Net over F₂₇ — Digital

Digital (33, 110, 220)-net over F₂₇, using

net from sequence [i] based on digital (33, 219)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 33 and N(F) ≥ 220, using
  - Drinfeld modules of rank 1 [i]

(33, 33+77, 7352)-Net in Base 27 — Upper bound on s

There is no (33, 110, 7353)-net in base 27, because

1 times m-reduction [i] would yield (33, 109, 7353)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 049143 245736 518317 052904 012833 558640 127468 592134 798930 591556 907518 665214 803065 873240 306267 442777 451052 610992 663804 297216 364837 623925 874953 640314 733220 263417 > 27¹⁰⁹ [i]