MinT - Best Known (95−11, 95, s)-Nets in Base 9

Best Known (95−11, 95, s)-Nets in Base 9

(95−11, 95, 3358680)-Net over F₉ — Constructive and digital

Digital (84, 95, 3358680)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 13, 3240)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹³, 3240, F₉, 5, 5) (dual of [(3240, 5), 16187, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(9¹³, 6481, F₉, 5) (dual of [6481, 6468, 6]-code), using
      - base reduction for projective spaces (embedding PG(6,81) in PG(12,9)) [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [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]

(95−11, 95, large)-Net over F₉ — Digital

Digital (84, 95, large)-net over F₉, using

t-expansion [i] based on digital (83, 95, large)-net over F₉, using
- 2 times m-reduction [i] based on digital (83, 97, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(95−11, 95, large)-Net in Base 9 — Upper bound on s

There is no (84, 95, large)-net in base 9, because

9 times m-reduction [i] would yield (84, 86, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]