MinT - Best Known (105−71, 105, s)-Nets in Base 27

Best Known (105−71, 105, s)-Nets in Base 27

(105−71, 105, 114)-Net over F₂₇ — Constructive and digital

Digital (34, 105, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 105, 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]

(105−71, 105, 172)-Net in Base 27 — Constructive

(34, 105, 172)-net in base 27, using

3 times m-reduction [i] based on (34, 108, 172)-net in base 27, using
- base change [i] based on digital (7, 81, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(105−71, 105, 220)-Net over F₂₇ — Digital

Digital (34, 105, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 105, 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]

(105−71, 105, 9564)-Net in Base 27 — Upper bound on s

There is no (34, 105, 9565)-net in base 27, because

1 times m-reduction [i] would yield (34, 104, 9565)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72932 868815 322751 161548 050341 281292 654739 256426 524984 761327 083883 497074 313918 733697 470974 974956 924123 349844 369628 511857 460822 012953 662984 830818 706227 > 27¹⁰⁴ [i]