MinT - Best Known (49, 49+55, s)-Nets in Base 27

Best Known (49, 49+55, s)-Nets in Base 27

(49, 49+55, 192)-Net over F₂₇ — Constructive and digital

Digital (49, 104, 192)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 38, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]
2. digital (11, 66, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(49, 49+55, 370)-Net in Base 27 — Constructive

(49, 104, 370)-net in base 27, using

t-expansion [i] based on (43, 104, 370)-net in base 27, using
- 4 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]

(49, 49+55, 457)-Net over F₂₇ — Digital

Digital (49, 104, 457)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+55, 121274)-Net in Base 27 — Upper bound on s

There is no (49, 104, 121275)-net in base 27, because

1 times m-reduction [i] would yield (49, 103, 121275)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 508161 164976 365451 791149 837586 693780 695505 624584 146191 242606 898780 285284 279330 106128 551406 478970 506703 990376 863371 024555 043185 603231 243893 147827 > 27¹⁰³ [i]