MinT - Best Known (50−29, 50, s)-Nets in Base 128

Best Known (50−29, 50, s)-Nets in Base 128

(50−29, 50, 384)-Net over F₁₂₈ — Constructive and digital

Digital (21, 50, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (21, 51, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 18, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]
  2. digital (3, 33, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(50−29, 50, 514)-Net in Base 128 — Constructive

(21, 50, 514)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 16, 257)-net in base 128, using
  - base change [i] based on digital (0, 14, 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]
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 (see above)

(50−29, 50, 530)-Net over F₁₂₈ — Digital

Digital (21, 50, 530)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁵⁰, 530, F₁₂₈, 29) (dual of [530, 480, 30]-code), using
- 138 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 12 times 0, 1, 44 times 0, 1, 77 times 0) [i] based on linear OA(128⁴⁴, 386, F₁₂₈, 29) (dual of [386, 342, 30]-code), using
  - extended algebraic-geometric code AG_e(F,356P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(50−29, 50, 1129500)-Net in Base 128 — Upper bound on s

There is no (21, 50, 1129501)-net in base 128, because

1 times m-reduction [i] would yield (21, 49, 1129501)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 17 918101 846414 177458 786129 303990 607225 935316 814356 393923 151331 845396 381880 908208 662307 778795 022125 844344 > 128⁴⁹ [i]