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

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

(108−51, 108, 210)-Net over F₂₇ — Constructive and digital

Digital (57, 108, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 21, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 29, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 58, 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]

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

(57, 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]

(108−51, 108, 951)-Net over F₂₇ — Digital

Digital (57, 108, 951)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−51, 108, 523454)-Net in Base 27 — Upper bound on s

There is no (57, 108, 523455)-net in base 27, because

1 times m-reduction [i] would yield (57, 107, 523455)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1431 992267 895991 611379 455385 834433 561202 733872 832791 081104 671098 495616 641093 500132 382376 858667 409381 109064 140346 545326 812544 420218 258023 229048 480120 066295 > 27¹⁰⁷ [i]