MinT - Best Known (120−21, 120, s)-Nets in Base 9

Best Known (120−21, 120, s)-Nets in Base 9

(120−21, 120, 53160)-Net over F₉ — Constructive and digital

Digital (99, 120, 53160)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (88, 109, 53144)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹⁰⁹, 53144, F₉, 21, 21) (dual of [(53144, 21), 1115915, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(9¹⁰⁹, 531441, F₉, 21) (dual of [531441, 531332, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 9⁶âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(120−21, 120, 551670)-Net over F₉ — Digital

Digital (99, 120, 551670)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(120−21, 120, large)-Net in Base 9 — Upper bound on s

There is no (99, 120, large)-net in base 9, because

19 times m-reduction [i] would yield (99, 101, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]