MinT - Best Known (51, 108, s)-Nets in Base 27

Best Known (51, 108, s)-Nets in Base 27

(51, 108, 192)-Net over F₂₇ — Constructive and digital

Digital (51, 108, 192)-net over F₂₇, using

1 times m-reduction [i] based on digital (51, 109, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 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, 69, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(51, 108, 370)-Net in Base 27 — Constructive

(51, 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]

(51, 108, 477)-Net over F₂₇ — Digital

Digital (51, 108, 477)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 108, 128184)-Net in Base 27 — Upper bound on s

There is no (51, 108, 128185)-net in base 27, because

1 times m-reduction [i] would yield (51, 107, 128185)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1431 981674 478252 989213 071724 599060 004424 364283 943828 228225 231290 011284 115379 353667 603430 375273 739096 630239 148551 545532 780564 735771 874188 983448 197211 961809 > 27¹⁰⁷ [i]