MinT - Best Known (89, 89+23, s)-Nets in Base 9

Best Known (89, 89+23, s)-Nets in Base 9

(89, 89+23, 5378)-Net over F₉ — Constructive and digital

Digital (89, 112, 5378)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (78, 101, 5368)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹⁰¹, 5368, F₉, 23, 23) (dual of [(5368, 23), 123363, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(9¹⁰¹, 59049, F₉, 23) (dual of [59049, 58948, 24]-code), using
      - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(89, 89+23, 81617)-Net over F₉ — Digital

Digital (89, 112, 81617)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+23, large)-Net in Base 9 — Upper bound on s

There is no (89, 112, large)-net in base 9, because

21 times m-reduction [i] would yield (89, 91, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]