MinT - Best Known (108−42, 108, s)-Nets in Base 27

Best Known (108−42, 108, s)-Nets in Base 27

(108−42, 108, 286)-Net over F₂₇ — Constructive and digital

Digital (66, 108, 286)-net over F₂₇, using

1 times m-reduction [i] based on digital (66, 109, 286)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 14, 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, 18, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (6, 27, 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]
  4. digital (7, 50, 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]

(108−42, 108, 730)-Net in Base 27 — Constructive

(66, 108, 730)-net in base 27, using

t-expansion [i] based on (63, 108, 730)-net in base 27, using
- base change [i] based on digital (36, 81, 730)-net over F₈₁, using
  - net from sequence [i] based on digital (36, 729)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 36 and N(F) ≥ 730, using
      - the Hermitian function field over F₈₁ [i]

(108−42, 108, 3679)-Net over F₂₇ — Digital

Digital (66, 108, 3679)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−42, 108, 7670399)-Net in Base 27 — Upper bound on s

There is no (66, 108, 7670400)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38662 283907 391188 646949 079441 650916 663387 175828 017308 975248 725281 606301 440396 642093 772976 873881 385224 284756 208005 458759 320280 122962 024930 730222 184076 427521 > 27¹⁰⁸ [i]