MinT - Best Known (42, 42+51, s)-Nets in Base 27

Best Known (42, 42+51, s)-Nets in Base 27

(42, 42+51, 176)-Net over F₂₇ — Constructive and digital

Digital (42, 93, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 32, 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 (10, 61, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(42, 42+51, 338)-Net over F₂₇ — Digital

Digital (42, 93, 338)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+51, 370)-Net in Base 27 — Constructive

(42, 93, 370)-net in base 27, using

11 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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]

(42, 42+51, 72442)-Net in Base 27 — Upper bound on s

There is no (42, 93, 72443)-net in base 27, because

1 times m-reduction [i] would yield (42, 92, 72443)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484775 928540 236130 702217 386628 669036 421706 920947 398124 194025 994506 843026 284994 206740 209276 123097 769061 282485 837854 622182 593522 375759 > 27⁹² [i]