MinT - Best Known (7, 13, s)-Nets in Base 16

Best Known (7, 13, s)-Nets in Base 16

(7, 13, 514)-Net over F₁₆ — Constructive and digital

Digital (7, 13, 514)-net over F₁₆, using

1 times m-reduction [i] based on digital (7, 14, 514)-net over F₁₆, using
- trace code for nets [i] based on digital (0, 7, 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]

(7, 13, 516)-Net over F₁₆ — Digital

Digital (7, 13, 516)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹³, 516, F₁₆, 6) (dual of [516, 503, 7]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(16¹², 514, F₁₆, 6) (dual of [514, 502, 7]-code), using
    - trace code [i] based on linear OA(256⁶, 257, F₂₅₆, 6) (dual of [257, 251, 7]-code or 257-arc in PG(5,256)), using
      - extended Reedâ€“Solomon code RS_e(251,256) [i]
      - algebraic-geometric code AG(F,125P) with degPÂ =Â 2 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using the rational function field F₂₅₆(x) [i]
      - algebraic-geometric code AG(F, Q+82P) with degQ = 4 and degPÂ =Â 3 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)
      - algebraic-geometric code AG(F,50P) with degPÂ =Â 5 [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257 (see above)
  2. linear OA(16¹², 515, F₁₆, 5) (dual of [515, 503, 6]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16¹²Â > V_b^sâˆ’1(kâˆ’1)Â = 145 596792 247936 [i]
  3. linear OA(16⁰, 1, F₁₆, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(7, 13, 20004)-Net in Base 16 — Upper bound on s

There is no (7, 13, 20005)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 4504 186279 028476 > 16¹³ [i]