MinT - Best Known (51, 51+28, s)-Nets in Base 16

Best Known (51, 51+28, s)-Nets in Base 16

(51, 51+28, 581)-Net over F₁₆ — Constructive and digital

Digital (51, 79, 581)-net over F₁₆, using

1 times m-reduction [i] based on digital (51, 80, 581)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 20, 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 (31, 60, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(51, 51+28, 594)-Net in Base 16 — Constructive

(51, 79, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (9, 23, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (9, 24, 80)-net in base 16, using
    - base change [i] based on digital (1, 16, 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 (28, 56, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 28, 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]

(51, 51+28, 2868)-Net over F₁₆ — Digital

Digital (51, 79, 2868)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁹, 2868, F₁₆, 28) (dual of [2868, 2789, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷⁹, 4096, F₁₆, 28) (dual of [4096, 4017, 29]-code), using
  - an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(51, 51+28, 2512130)-Net in Base 16 — Upper bound on s

There is no (51, 79, 2512131)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133499 372611 946218 306968 697397 846939 566217 340430 638164 819840 003148 881116 579087 079582 549701 164736 > 16⁷⁹ [i]