MinT - Best Known (78, 78+31, s)-Nets in Base 9

Best Known (78, 78+31, s)-Nets in Base 9

(78, 78+31, 750)-Net over F₉ — Constructive and digital

Digital (78, 109, 750)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (63, 94, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 47, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(78, 78+31, 5204)-Net over F₉ — Digital

Digital (78, 109, 5204)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁰⁹, 5204, F₉, 31) (dual of [5204, 5095, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹⁰⁹, 6561, F₉, 31) (dual of [6561, 6452, 32]-code), using
  - an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(78, 78+31, 5959668)-Net in Base 9 — Upper bound on s

There is no (78, 109, 5959669)-net in base 9, because

1 times m-reduction [i] would yield (78, 108, 5959669)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 433836 832765 748160 693913 798971 198927 379404 354357 165503 576051 415626 878554 103341 290880 393920 131976 771737 > 9¹⁰⁸ [i]