MinT - Best Known (24, 24+34, s)-Nets in Base 128

Best Known (24, 24+34, s)-Nets in Base 128

(24, 24+34, 384)-Net over F₁₂₈ — Constructive and digital

Digital (24, 58, 384)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (24, 60, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 21, 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, 39, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(24, 24+34, 408)-Net in Base 128 — Constructive

(24, 58, 408)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 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]
2. (6, 40, 258)-net in base 128, using
  - base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(24, 24+34, 541)-Net over F₁₂₈ — Digital

Digital (24, 58, 541)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁵⁸, 541, F₁₂₈, 34) (dual of [541, 483, 35]-code), using
- 146 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 44 times 0, 1, 66 times 0) [i] based on linear OA(128⁴⁹, 386, F₁₂₈, 34) (dual of [386, 337, 35]-code), using
  - extended algebraic-geometric code AG_e(F,351P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(24, 24+34, 873860)-Net in Base 128 — Upper bound on s

There is no (24, 58, 873861)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 165 266008 267600 098579 577143 743466 165988 568317 464986 730504 557165 378564 320269 995044 965348 511907 173172 634425 911348 044001 435744 > 128⁵⁸ [i]