MinT - Best Known (20, 48, s)-Nets in Base 128

Best Known (20, 48, s)-Nets in Base 128

(20, 48, 384)-Net over F₁₂₈ — Constructive and digital

Digital (20, 48, 384)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 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, 31, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(20, 48, 494)-Net over F₁₂₈ — Digital

Digital (20, 48, 494)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(128⁴⁸, 494, F₁₂₈, 28) (dual of [494, 446, 29]-code), using
- 103 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 6 times 0, 1, 27 times 0, 1, 67 times 0) [i] based on linear OA(128⁴³, 386, F₁₂₈, 28) (dual of [386, 343, 29]-code), using
  - extended algebraic-geometric code AG_e(F,357P) [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(20, 48, 514)-Net in Base 128 — Constructive

(20, 48, 514)-net in base 128, using

base change [i] based on digital (14, 42, 514)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. 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. digital (0, 28, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(20, 48, 798675)-Net in Base 128 — Upper bound on s

There is no (20, 48, 798676)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 139985 464223 345368 217791 555835 186749 173412 360277 943927 117873 836152 125998 790470 546621 314895 615956 217244 > 128⁴⁸ [i]