MinT - Best Known (122−14, 122, s)-Nets in Base 9

Best Known (122−14, 122, s)-Nets in Base 9

(122−14, 122, 2396942)-Net over F₉ — Constructive and digital

Digital (108, 122, 2396942)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (9, 16, 200)-net over F₉, using
  - trace code for nets [i] based on digital (1, 8, 100)-net over F₈₁, using
    - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
2. digital (92, 106, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵³, 1198371, F₈₁, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(81⁵³, 8388597, F₈₁, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵³, large, F₈₁, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(122−14, 122, large)-Net over F₉ — Digital

Digital (108, 122, large)-net over F₉, using

t-expansion [i] based on digital (107, 122, large)-net over F₉, using
- 3 times m-reduction [i] based on digital (107, 125, large)-net over F₉, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(122−14, 122, large)-Net in Base 9 — Upper bound on s

There is no (108, 122, large)-net in base 9, because

12 times m-reduction [i] would yield (108, 110, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]