MinT - Best Known (34, 34+75, s)-Nets in Base 27

Best Known (34, 34+75, s)-Nets in Base 27

(34, 34+75, 114)-Net over F₂₇ — Constructive and digital

Digital (34, 109, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 109, 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]

(34, 34+75, 160)-Net in Base 27 — Constructive

(34, 109, 160)-net in base 27, using

27¹ times duplication [i] based on (33, 108, 160)-net in base 27, using
- t-expansion [i] based on (32, 108, 160)-net in base 27, using
  - base change [i] based on digital (5, 81, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(34, 34+75, 220)-Net over F₂₇ — Digital

Digital (34, 109, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 109, 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]

(34, 34+75, 8471)-Net in Base 27 — Upper bound on s

There is no (34, 109, 8472)-net in base 27, because

1 times m-reduction [i] would yield (34, 108, 8472)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38684 423510 370284 476866 588697 295408 253767 629957 313294 047798 542350 617926 443890 333822 172542 726936 839102 456383 052943 490468 024223 096200 868441 789786 642972 031985 > 27¹⁰⁸ [i]