MinT - Best Known (37, 37+21, s)-Nets in Base 16

Best Known (37, 37+21, s)-Nets in Base 16

(37, 37+21, 579)-Net over F₁₆ — Constructive and digital

Digital (37, 58, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 16, 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 (21, 42, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 21, 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]

(37, 37+21, 2156)-Net over F₁₆ — Digital

Digital (37, 58, 2156)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁸, 2156, F₁₆, 21) (dual of [2156, 2098, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁵⁸, 4096, F₁₆, 21) (dual of [4096, 4038, 22]-code), using
  - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(37, 37+21, 2204793)-Net in Base 16 — Upper bound on s

There is no (37, 58, 2204794)-net in base 16, because

1 times m-reduction [i] would yield (37, 57, 2204794)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 431 360047 631786 415557 979010 378428 776398 834485 869232 151158 539270 295976 > 16⁵⁷ [i]