MinT - Best Known (114−48, 114, s)-Nets in Base 9

Best Known (114−48, 114, s)-Nets in Base 9

(114−48, 114, 344)-Net over F₉ — Constructive and digital

Digital (66, 114, 344)-net over F₉, using

4 times m-reduction [i] based on digital (66, 118, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 59, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(114−48, 114, 488)-Net over F₉ — Digital

Digital (66, 114, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 57, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(114−48, 114, 41761)-Net in Base 9 — Upper bound on s

There is no (66, 114, 41762)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 078870 604334 289218 862450 316162 216167 937885 847842 681883 184808 765275 212312 841685 010841 538163 018762 049498 716289 > 9¹¹⁴ [i]