MinT - Best Known (124−15, 124, s)-Nets in Base 9

Best Known (124−15, 124, s)-Nets in Base 9

(124−15, 124, 2396770)-Net over F₉ — Constructive and digital

Digital (109, 124, 2396770)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (99, 114, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (42, 57, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵⁷, 1198371, F₈₁, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(81⁵⁷, 8388598, F₈₁, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵⁷, large, F₈₁, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 81⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(124−15, 124, large)-Net over F₉ — Digital

Digital (109, 124, large)-net over F₉, using

t-expansion [i] based on digital (107, 124, large)-net over F₉, using
- 1 times m-reduction [i] based on digital (107, 125, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(124−15, 124, large)-Net in Base 9 — Upper bound on s

There is no (109, 124, large)-net in base 9, because

13 times m-reduction [i] would yield (109, 111, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]