MinT - Best Known (108−75, 108, s)-Nets in Base 27

Best Known (108−75, 108, s)-Nets in Base 27

(108−75, 108, 114)-Net over F₂₇ — Constructive and digital

Digital (33, 108, 114)-net over F₂₇, using

t-expansion [i] based on digital (23, 108, 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]

(108−75, 108, 160)-Net in Base 27 — Constructive

(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]

(108−75, 108, 220)-Net over F₂₇ — Digital

Digital (33, 108, 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]

(108−75, 108, 7748)-Net in Base 27 — Upper bound on s

There is no (33, 108, 7749)-net in base 27, because

1 times m-reduction [i] would yield (33, 107, 7749)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1437 377796 337556 741485 008396 116533 382296 310457 512301 206151 955016 256306 745654 150572 727395 757150 894387 949586 531071 187180 994513 373141 548513 514489 647056 648179 > 27¹⁰⁷ [i]