MinT - Best Known (82, 82+13, s)-Nets in Base 9

Best Known (82, 82+13, s)-Nets in Base 9

(82, 82+13, 1398110)-Net over F₉ — Constructive and digital

Digital (82, 95, 1398110)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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 (76, 89, 1398100)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁸⁹, 1398100, F₉, 13, 13) (dual of [(1398100, 13), 18175211, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(9⁸⁹, 8388601, F₉, 13) (dual of [8388601, 8388512, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁸⁹, large, F₉, 13) (dual of [large, large−89, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(82, 82+13, large)-Net over F₉ — Digital

Digital (82, 95, large)-net over F₉, using

9⁶ times duplication [i] based on digital (76, 89, large)-net over F₉, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, large, F₉, 13) (dual of [large, large−89, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(82, 82+13, large)-Net in Base 9 — Upper bound on s

There is no (82, 95, large)-net in base 9, because

11 times m-reduction [i] would yield (82, 84, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]