MinT - Best Known (64−34, 64, s)-Nets in Base 9

Best Known (64−34, 64, s)-Nets in Base 9

(64−34, 64, 78)-Net over F₉ — Constructive and digital

Digital (30, 64, 78)-net over F₉, using

t-expansion [i] based on digital (22, 64, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(64−34, 64, 84)-Net in Base 9 — Constructive

(30, 64, 84)-net in base 9, using

2 times m-reduction [i] based on (30, 66, 84)-net in base 9, using
- base change [i] based on digital (8, 44, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(64−34, 64, 112)-Net over F₉ — Digital

Digital (30, 64, 112)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9⁶⁴, 112, F₉, 3, 34) (dual of [(112, 3), 272, 35]-NRT-code), using
- constructionÂ X applied to AG(3;F,292P) âŠ‚ AG(3;F,297P) [i] based on
  1. linear OOA(9⁶⁰, 109, F₉, 3, 34) (dual of [(109, 3), 267, 35]-NRT-code), using algebraic-geometric NRT-code AG(3;F,292P) [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]
  2. linear OOA(9⁵⁵, 109, F₉, 3, 29) (dual of [(109, 3), 272, 30]-NRT-code), using algebraic-geometric NRT-code AG(3;F,297P) [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110 (see above)
  3. linear OOA(9⁴, 3, F₉, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(9⁴, 9, F₉, 3, 4) (dual of [(9, 3), 23, 5]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;23,9) [i]

(64−34, 64, 3499)-Net in Base 9 — Upper bound on s

There is no (30, 64, 3500)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 801209 929110 349617 730512 519406 269530 651140 653703 000814 632801 > 9⁶⁴ [i]