MinT - Best Known (44, 44+63, s)-Nets in Base 27

Best Known (44, 44+63, s)-Nets in Base 27

(44, 44+63, 158)-Net over F₂₇ — Constructive and digital

Digital (44, 107, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 37, 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, 70, 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]

(44, 44+63, 280)-Net over F₂₇ — Digital

Digital (44, 107, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 107, 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]

(44, 44+63, 370)-Net in Base 27 — Constructive

(44, 107, 370)-net in base 27, using

t-expansion [i] based on (43, 107, 370)-net in base 27, using
- 1 times m-reduction [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]

(44, 44+63, 37429)-Net in Base 27 — Upper bound on s

There is no (44, 107, 37430)-net in base 27, because

1 times m-reduction [i] would yield (44, 106, 37430)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 062660 227944 903213 142840 598359 750608 598147 878517 248526 205581 983038 991340 570499 228002 432455 954329 876927 324118 046074 418562 053726 843799 078744 357316 245113 > 27¹⁰⁶ [i]