MinT - Best Known (110−65, 110, s)-Nets in Base 27

Best Known (110−65, 110, s)-Nets in Base 27

(110−65, 110, 158)-Net over F₂₇ — Constructive and digital

Digital (45, 110, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 38, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 72, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(110−65, 110, 280)-Net over F₂₇ — Digital

Digital (45, 110, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 110, 280)-net over F₂₇, using
- net from sequence [i] based on digital (42, 279)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
    - Drinfeld modules of rank 1 [i]

(110−65, 110, 370)-Net in Base 27 — Constructive

(45, 110, 370)-net in base 27, using

27² times duplication [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(110−65, 110, 36922)-Net in Base 27 — Upper bound on s

There is no (45, 110, 36923)-net in base 27, because

1 times m-reduction [i] would yield (45, 109, 36923)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 044176 652236 661049 770665 461066 984177 083263 496593 696449 401560 013465 500470 835712 137308 737390 091706 555495 873607 110423 870172 417481 141342 819326 868698 022445 382849 > 27¹⁰⁹ [i]