MinT - Best Known (84−38, 84, s)-Nets in Base 27

Best Known (84−38, 84, s)-Nets in Base 27

(84−38, 84, 210)-Net over F₂₇ — Constructive and digital

Digital (46, 84, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 16, 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, 23, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 45, 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]

(84−38, 84, 370)-Net in Base 27 — Constructive

(46, 84, 370)-net in base 27, using

t-expansion [i] based on (43, 84, 370)-net in base 27, using
- 24 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]

(84−38, 84, 1020)-Net over F₂₇ — Digital

Digital (46, 84, 1020)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−38, 84, 649187)-Net in Base 27 — Upper bound on s

There is no (46, 84, 649188)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 716176 136680 955565 743893 395561 203380 328406 695741 508536 954293 755545 027914 945678 753614 711980 391300 987455 174458 287726 835857 > 27⁸⁴ [i]