MinT - Best Known (23, 56, s)-Nets in Base 128

Best Known (23, 56, s)-Nets in Base 128

(23, 56, 384)-Net over F₁₂₈ — Constructive and digital

Digital (23, 56, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (23, 57, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 20, 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, 37, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(23, 56, 508)-Net over F₁₂₈ — Digital

Digital (23, 56, 508)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁵⁶, 508, F₁₂₈, 33) (dual of [508, 452, 34]-code), using
- 114 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 0, 0, 1, 13 times 0, 1, 34 times 0, 1, 60 times 0) [i] based on linear OA(128⁴⁸, 386, F₁₂₈, 33) (dual of [386, 338, 34]-code), using
  - extended algebraic-geometric code AG_e(F,352P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(23, 56, 514)-Net in Base 128 — Constructive

(23, 56, 514)-net in base 128, using

base change [i] based on digital (16, 49, 514)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 16, 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 (0, 33, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(23, 56, 938135)-Net in Base 128 — Upper bound on s

There is no (23, 56, 938136)-net in base 128, because

1 times m-reduction [i] would yield (23, 55, 938136)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 78 804180 396429 832080 647930 977895 905247 707314 594071 105344 968802 620596 243517 043276 853831 911070 144302 750597 647112 970653 > 128⁵⁵ [i]