MinT - Best Known (108−63, 108, s)-Nets in Base 27

Best Known (108−63, 108, s)-Nets in Base 27

(108−63, 108, 164)-Net over F₂₇ — Constructive and digital

Digital (45, 108, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 38, 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]
2. digital (7, 70, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(108−63, 108, 280)-Net over F₂₇ — Digital

Digital (45, 108, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 108, 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]

(108−63, 108, 370)-Net in Base 27 — Constructive

(45, 108, 370)-net in base 27, using

t-expansion [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]

(108−63, 108, 41629)-Net in Base 27 — Upper bound on s

There is no (45, 108, 41630)-net in base 27, because

1 times m-reduction [i] would yield (45, 107, 41630)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1432 141303 983423 726876 281653 429580 838361 422225 929364 084500 367956 283804 363579 681406 649023 609506 039179 724994 537808 827828 103188 568584 982083 399855 124870 006809 > 27¹⁰⁷ [i]