MinT - Best Known (139−28, 139, s)-Nets in Base 9

Best Known (139−28, 139, s)-Nets in Base 9

(139−28, 139, 4247)-Net over F₉ — Constructive and digital

Digital (111, 139, 4247)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (93, 121, 4217)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²¹, 4217, F₉, 28, 28) (dual of [(4217, 28), 117955, 29]-NRT-code), using
    - OA 14-folding and stacking [i] based on linear OA(9¹²¹, 59038, F₉, 28) (dual of [59038, 58917, 29]-code), using
      - discarding factors / shortening the dual code 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]

(139−28, 139, 111674)-Net over F₉ — Digital

Digital (111, 139, 111674)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−28, 139, large)-Net in Base 9 — Upper bound on s

There is no (111, 139, large)-net in base 9, because

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