MinT - Best Known (11−7, 11, s)-Nets in Base 128

Best Known (11−7, 11, s)-Nets in Base 128

(11−7, 11, 279)-Net over F₁₂₈ — Constructive and digital

Digital (4, 11, 279)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 8, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(11−7, 11, 335)-Net over F₁₂₈ — Digital

Digital (4, 11, 335)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128¹¹, 335, F₁₂₈, 7) (dual of [335, 324, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(128¹¹, 381, F₁₂₈, 7) (dual of [381, 370, 8]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 381Â | 128²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(11−7, 11, 386)-Net in Base 128 — Constructive

(4, 11, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. (1, 8, 257)-net in base 128, using
  - base change [i] based on digital (0, 7, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(11−7, 11, 151220)-Net in Base 128 — Upper bound on s

There is no (4, 11, 151221)-net in base 128, because

1 times m-reduction [i] would yield (4, 10, 151221)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1180 606170 292646 370092 > 128¹⁰ [i]