MinT - Best Known (109−39, 109, s)-Nets in Base 16

Best Known (109−39, 109, s)-Nets in Base 16

(109−39, 109, 585)-Net over F₁₆ — Constructive and digital

Digital (70, 109, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 25, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (45, 84, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 42, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(109−39, 109, 594)-Net in Base 16 — Constructive

(70, 109, 594)-net in base 16, using

16¹ times duplication [i] based on (69, 108, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (11, 30, 80)-net in base 16, using
    - base change [i] based on digital (1, 20, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (39, 78, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 39, 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]

(109−39, 109, 3176)-Net over F₁₆ — Digital

Digital (70, 109, 3176)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰⁹, 3176, F₁₆, 39) (dual of [3176, 3067, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹⁰⁹, 4096, F₁₆, 39) (dual of [4096, 3987, 40]-code), using
  - an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]

(109−39, 109, 3694830)-Net in Base 16 — Upper bound on s

There is no (70, 109, 3694831)-net in base 16, because

1 times m-reduction [i] would yield (70, 108, 3694831)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11090 706825 764878 699463 525117 729868 815712 201128 834851 682539 998222 764595 255606 271002 304239 069433 118424 054614 400828 361829 296330 618936 > 16¹⁰⁸ [i]