MinT - Best Known (79−42, 79, s)-Nets in Base 9

Best Known (79−42, 79, s)-Nets in Base 9

(79−42, 79, 92)-Net over F₉ — Constructive and digital

Digital (37, 79, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (13, 55, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(79−42, 79, 94)-Net in Base 9 — Constructive

(37, 79, 94)-net in base 9, using

2 times m-reduction [i] based on (37, 81, 94)-net in base 9, using
- base change [i] based on digital (10, 54, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(79−42, 79, 130)-Net over F₉ — Digital

Digital (37, 79, 130)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9⁷⁹, 130, F₉, 3, 42) (dual of [(130, 3), 311, 43]-NRT-code), using
- constructionÂ X applied to AG(3;F,338P) âŠ‚ AG(3;F,343P) [i] based on
  1. linear OOA(9⁷⁵, 127, F₉, 3, 42) (dual of [(127, 3), 306, 43]-NRT-code), using algebraic-geometric NRT-code AG(3;F,338P) [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]
  2. linear OOA(9⁷⁰, 127, F₉, 3, 37) (dual of [(127, 3), 311, 38]-NRT-code), using algebraic-geometric NRT-code AG(3;F,343P) [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128 (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]

(79−42, 79, 4206)-Net in Base 9 — Upper bound on s

There is no (37, 79, 4207)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2439 339888 779912 482130 425470 316549 058920 190732 437088 339335 479316 256954 100953 > 9⁷⁹ [i]