MinT - Best Known (105−11, 105, s)-Nets in Base 9

Best Known (105−11, 105, s)-Nets in Base 9

(105−11, 105, 3413769)-Net over F₉ — Constructive and digital

Digital (94, 105, 3413769)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (18, 23, 58329)-net over F₉, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 6481)-net over F₉, using
      - s-reduction based on digital (0, 0, s)-net over F₉ with arbitrarily large s, using
        arbitrary set of 9⁰ points [i]
    2. digital (0, 0, 6481)-net over F₉ (see above)
    3. digital (0, 0, 6481)-net over F₉ (see above)
    4. digital (0, 0, 6481)-net over F₉ (see above)
    5. digital (0, 1, 6481)-net over F₉, using
      - s-reduction based on digital (0, 1, s)-net over F₉ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    6. digital (0, 1, 6481)-net over F₉ (see above)
    7. digital (0, 1, 6481)-net over F₉ (see above)
    8. digital (3, 5, 6481)-net over F₉, using
      - s-reduction based on digital (3, 5, 7381)-net over F₉, using
        kÂ =Â 2 construction [i]
    9. digital (10, 15, 6481)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9¹⁵, 6481, F₉, 5, 5) (dual of [(6481, 5), 32390, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(9¹⁵, 12963, F₉, 5) (dual of [12963, 12948, 6]-code), using
        1 times code embedding in larger space [i] based on linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
        trace code [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]

(105−11, 105, large)-Net over F₉ — Digital

Digital (94, 105, large)-net over F₉, using

9¹ times duplication [i] based on digital (93, 104, large)-net over F₉, using
- t-expansion [i] based on digital (89, 104, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(105−11, 105, large)-Net in Base 9 — Upper bound on s

There is no (94, 105, large)-net in base 9, because

9 times m-reduction [i] would yield (94, 96, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (105−11, 105, s)-Nets in Base 9

(105−11, 105, 3413769)-Net over F9 — Constructive and digital

(105−11, 105, large)-Net over F9 — Digital

(105−11, 105, large)-Net in Base 9 — Upper bound on s

(105−11, 105, 3413769)-Net over F₉ — Constructive and digital

(105−11, 105, large)-Net over F₉ — Digital