MinT - Best Known (129−37, 129, s)-Nets in Base 9

Best Known (129−37, 129, s)-Nets in Base 9

(129−37, 129, 772)-Net over F₉ — Constructive and digital

Digital (92, 129, 772)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 23, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]
2. digital (69, 106, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 53, 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]

(129−37, 129, 5349)-Net over F₉ — Digital

Digital (92, 129, 5349)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹²⁹, 5349, F₉, 37) (dual of [5349, 5220, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹²⁹, 6561, F₉, 37) (dual of [6561, 6432, 38]-code), using
  - an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 9⁴âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]

(129−37, 129, 5764528)-Net in Base 9 — Upper bound on s

There is no (92, 129, 5764529)-net in base 9, because

1 times m-reduction [i] would yield (92, 128, 5764529)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 139 008780 756295 412351 128419 105206 964704 142304 154328 156895 133067 320577 623906 592421 676907 794813 592239 297710 511022 969577 302993 > 9¹²⁸ [i]