MinT - Best Known (105, 116, s)-Nets in Base 9

Best Known (105, 116, s)-Nets in Base 9

(105, 116, 6710880)-Net over F₉ — Constructive and digital

Digital (105, 116, 6710880)-net over F₉, using

9¹ times duplication [i] based on digital (104, 115, 6710880)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (28, 33, 4194301)-net over F₉, using
    - net defined by OOA [i] based on linear OOA(9³³, 4194301, F₉, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(9³³, large, F₉, 5) (dual of [large, large−33, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 9¹⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
  2. digital (71, 82, 3355440)-net over F₉, using
    - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₈₁, using
      - net defined by OOA [i] based on linear OOA(81⁴¹, 1677720, F₈₁, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(81⁴¹, 8388601, F₈₁, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴¹, large, F₈₁, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(105, 116, large)-Net over F₉ — Digital

Digital (105, 116, large)-net over F₉, using

t-expansion [i] based on digital (101, 116, large)-net over F₉, using
- 2 times m-reduction [i] based on digital (101, 118, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(105, 116, large)-Net in Base 9 — Upper bound on s

There is no (105, 116, large)-net in base 9, because

9 times m-reduction [i] would yield (105, 107, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]