MinT - Best Known (30, 30+43, s)-Nets in Base 128

Best Known (30, 30+43, s)-Nets in Base 128

(30, 30+43, 417)-Net over F₁₂₈ — Constructive and digital

Digital (30, 73, 417)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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 (9, 52, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(30, 30+43, 432)-Net in Base 128 — Constructive

(30, 73, 432)-net in base 128, using

128¹ times duplication [i] based on (29, 72, 432)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (3, 24, 257)-net in base 128, using
    - base change [i] based on digital (0, 21, 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. digital (5, 48, 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]

(30, 30+43, 620)-Net over F₁₂₈ — Digital

Digital (30, 73, 620)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁷³, 620, F₁₂₈, 43) (dual of [620, 547, 44]-code), using
- 40 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 39 times 0) [i] based on linear OA(128⁷², 579, F₁₂₈, 43) (dual of [579, 507, 44]-code), using
  - constructionÂ X applied to AG(F,532P) âŠ‚ AG(F,534P) [i] based on
    1. linear OA(128⁷¹, 576, F₁₂₈, 43) (dual of [576, 505, 44]-code), using algebraic-geometric code AG(F,532P) [i] based on function field F/F₁₂₈ with g(F) = 28 and N(F) ≥ 577, using
      - a curve from the manYPoints database [i]
    2. linear OA(128⁶⁹, 576, F₁₂₈, 41) (dual of [576, 507, 42]-code), using algebraic-geometric code AG(F,534P) [i] based on function field F/F₁₂₈ with g(F) = 28 and N(F) ≥ 577 (see above)
    3. linear OA(128¹, 3, F₁₂₈, 1) (dual of [3, 2, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(30, 30+43, 1146581)-Net in Base 128 — Upper bound on s

There is no (30, 73, 1146582)-net in base 128, because

1 times m-reduction [i] would yield (30, 72, 1146582)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 52 374826 424327 930355 244725 198378 367802 228260 608341 969260 965729 342921 167176 873368 639793 754383 163881 560996 644381 343975 543754 696030 787364 532292 299882 477195 > 128⁷² [i]