MinT - Best Known (72−39, 72, s)-Nets in Base 8

Best Known (72−39, 72, s)-Nets in Base 8

(72−39, 72, 74)-Net over F₈ — Constructive and digital

Digital (33, 72, 74)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (14, 53, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(72−39, 72, 99)-Net over F₈ — Digital

Digital (33, 72, 99)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁷², 99, F₈, 3, 39) (dual of [(99, 3), 225, 40]-NRT-code), using
- constructionÂ X applied to AG(3;F,245P) âŠ‚ AG(3;F,252P) [i] based on
  1. linear OOA(8⁶⁶, 95, F₈, 3, 39) (dual of [(95, 3), 219, 40]-NRT-code), using algebraic-geometric NRT-code AG(3;F,245P) [i] based on function field F/F₈ with g(F) = 27 and N(F) ≥ 96, using
    - a curve from the manYPoints database [i]
  2. linear OOA(8⁵⁹, 95, F₈, 3, 32) (dual of [(95, 3), 226, 33]-NRT-code), using algebraic-geometric NRT-code AG(3;F,252P) [i] based on function field F/F₈ with g(F) = 27 and N(F) ≥ 96 (see above)
  3. linear OOA(8⁶, 4, F₈, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8⁶, 8, F₈, 3, 6) (dual of [(8, 3), 18, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;18,8) [i]

(72−39, 72, 2672)-Net in Base 8 — Upper bound on s

There is no (33, 72, 2673)-net in base 8, because

1 times m-reduction [i] would yield (33, 71, 2673)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13206 340540 856231 790445 337708 425431 475347 714090 362673 836925 009184 > 8⁷¹ [i]