MinT - Best Known (138−27, 138, s)-Nets in Base 9

Best Known (138−27, 138, s)-Nets in Base 9

(138−27, 138, 4574)-Net over F₉ — Constructive and digital

Digital (111, 138, 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 (93, 120, 4542)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²⁰, 4542, F₉, 27, 27) (dual of [(4542, 27), 122514, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(9¹²⁰, 59047, F₉, 27) (dual of [59047, 58927, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹²⁰, 59048, F₉, 27) (dual of [59048, 58928, 28]-code), using
        1 times truncation [i] based on linear OA(9¹²¹, 59049, F₉, 28) (dual of [59049, 58928, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(138−27, 138, 153132)-Net over F₉ — Digital

Digital (111, 138, 153132)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−27, 138, large)-Net in Base 9 — Upper bound on s

There is no (111, 138, large)-net in base 9, because

25 times m-reduction [i] would yield (111, 113, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]