MinT - Best Known (43−25, 43, s)-Nets in Base 128

Best Known (43−25, 43, s)-Nets in Base 128

(43−25, 43, 384)-Net over F₁₂₈ — Constructive and digital

Digital (18, 43, 384)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 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, 28, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(43−25, 43, 473)-Net over F₁₂₈ — Digital

Digital (18, 43, 473)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁴³, 473, F₁₂₈, 25) (dual of [473, 430, 26]-code), using
- 200 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 32 times 0, 1, 57 times 0, 1, 81 times 0) [i] based on linear OA(128³², 262, F₁₂₈, 25) (dual of [262, 230, 26]-code), using
  - extended algebraic-geometric code AG_e(F,236P) [i] based on function field F/F₁₂₈ with g(F) = 7 and N(F) ≥ 262, using
    - a curve from the manYPoints database [i]

(43−25, 43, 514)-Net in Base 128 — Constructive

(18, 43, 514)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (2, 14, 257)-net in base 128, using
  - 2 times m-reduction [i] based on (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. (4, 29, 257)-net in base 128, using
  - 3 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
    - base change [i] based on digital (0, 28, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(43−25, 43, 988075)-Net in Base 128 — Upper bound on s

There is no (18, 43, 988076)-net in base 128, because

1 times m-reduction [i] would yield (18, 42, 988076)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 31828 883295 805807 618039 649716 172371 639535 857759 503098 807060 850490 144502 875413 322788 926616 > 128⁴² [i]