MinT - Best Known (77, 77+11, s)-Nets in Base 9

Best Known (77, 77+11, s)-Nets in Base 9

(77, 77+11, 3355456)-Net over F₉ — Constructive and digital

Digital (77, 88, 3355456)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [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]

(77, 77+11, large)-Net over F₉ — Digital

Digital (77, 88, large)-net over F₉, using

t-expansion [i] based on digital (76, 88, large)-net over F₉, using
- 1 times m-reduction [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]

(77, 77+11, large)-Net in Base 9 — Upper bound on s

There is no (77, 88, large)-net in base 9, because

9 times m-reduction [i] would yield (77, 79, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]