MinT - Best Known (19, 48, s)-Nets in Base 128

Best Known (19, 48, s)-Nets in Base 128

(19, 48, 345)-Net over F₁₂₈ — Constructive and digital

Digital (19, 48, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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 (5, 34, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(19, 48, 386)-Net in Base 128 — Constructive

(19, 48, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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. (5, 34, 257)-net in base 128, using
  - 6 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
    - base change [i] based on digital (0, 35, 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]

(19, 48, 389)-Net over F₁₂₈ — Digital

Digital (19, 48, 389)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁴⁸, 389, F₁₂₈, 29) (dual of [389, 341, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(128⁴⁸, 394, F₁₂₈, 29) (dual of [394, 346, 30]-code), using
  - constructionÂ X applied to AG(F,355P) âŠ‚ AG(F,360P) [i] based on
    1. linear OA(128⁴⁴, 385, F₁₂₈, 29) (dual of [385, 341, 30]-code), using algebraic-geometric code AG(F,355P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
      - a curve from the manYPoints database [i]
    2. linear OA(128³⁹, 385, F₁₂₈, 24) (dual of [385, 346, 25]-code), using algebraic-geometric code AG(F,360P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386 (see above)
    3. linear OA(128⁴, 9, F₁₂₈, 4) (dual of [9, 5, 5]-code or 9-arc in PG(3,128)), using
      - discarding factors / shortening the dual code based on linear OA(128⁴, 128, F₁₂₈, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
        Reedâ€“Solomon code RS(124,128) [i]

(19, 48, 513)-Net in Base 128

(19, 48, 513)-net in base 128, using

t-expansion [i] based on (17, 48, 513)-net in base 128, using
- 24 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 513)-net over F₂₅₆, using
    - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(19, 48, 564746)-Net in Base 128 — Upper bound on s

There is no (19, 48, 564747)-net in base 128, because

1 times m-reduction [i] would yield (19, 47, 564747)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1093 628862 205742 286015 965660 096278 157306 182612 747003 826947 377749 355405 290087 022304 228797 200261 407872 > 128⁴⁷ [i]