MinT - Best Known (55, 55+30, s)-Nets in Base 16

Best Known (55, 55+30, s)-Nets in Base 16

(55, 55+30, 583)-Net over F₁₆ — Constructive and digital

Digital (55, 85, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 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 (34, 64, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 32, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(55, 55+30, 594)-Net in Base 16 — Constructive

(55, 85, 594)-net in base 16, using

1 times m-reduction [i] based on (55, 86, 594)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (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 (31, 62, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 31, 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]

(55, 55+30, 3071)-Net over F₁₆ — Digital

Digital (55, 85, 3071)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁵, 3071, F₁₆, 30) (dual of [3071, 2986, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸⁵, 4096, F₁₆, 30) (dual of [4096, 4011, 31]-code), using
  - an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(55, 55+30, 2851154)-Net in Base 16 — Upper bound on s

There is no (55, 85, 2851155)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 239752 200571 748967 657406 423466 824279 583370 335653 675723 288286 965245 961907 911203 817246 555563 309208 567376 > 16⁸⁵ [i]