MinT - Best Known (134−26, 134, s)-Nets in Base 9

Best Known (134−26, 134, s)-Nets in Base 9

(134−26, 134, 4574)-Net over F₉ — Constructive and digital

Digital (108, 134, 4574)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 18, 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 (90, 116, 4542)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹¹⁶, 4542, F₉, 26, 26) (dual of [(4542, 26), 117976, 27]-NRT-code), using
    - OA 13-folding and stacking [i] based on linear OA(9¹¹⁶, 59046, F₉, 26) (dual of [59046, 58930, 27]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹¹⁶, 59049, F₉, 26) (dual of [59049, 58933, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(134−26, 134, 165695)-Net over F₉ — Digital

Digital (108, 134, 165695)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(134−26, 134, large)-Net in Base 9 — Upper bound on s

There is no (108, 134, large)-net in base 9, because

24 times m-reduction [i] would yield (108, 110, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]