MinT - Best Known (108−13, 108, s)-Nets in Base 9

Best Known (108−13, 108, s)-Nets in Base 9

(108−13, 108, 2796245)-Net over F₉ — Constructive and digital

Digital (95, 108, 2796245)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 45)-net over F₉, using
  - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) [i]
2. digital (85, 98, 2796200)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁹⁸, 2796200, F₉, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁸, 8388601, F₉, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁹⁸, 8388602, F₉, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
        trace code [i] based on linear OOA(81⁴⁹, 4194301, F₈₁, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(81⁴⁹, 8388602, F₈₁, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁹, large, F₈₁, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(108−13, 108, large)-Net over F₉ — Digital

Digital (95, 108, large)-net over F₉, using

3 times m-reduction [i] based on digital (95, 111, large)-net over F₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(108−13, 108, large)-Net in Base 9 — Upper bound on s

There is no (95, 108, large)-net in base 9, because

11 times m-reduction [i] would yield (95, 97, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]