MinT - Best Known (53, 53+29, s)-Nets in Base 16

Best Known (53, 53+29, s)-Nets in Base 16

(53, 53+29, 583)-Net over F₁₆ — Constructive and digital

Digital (53, 82, 583)-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 (33, 62, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 31, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(53, 53+29, 594)-Net in Base 16 — Constructive

(53, 82, 594)-net in base 16, using

16¹ times duplication [i] based on (52, 81, 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 (29, 58, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 29, 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]

(53, 53+29, 2969)-Net over F₁₆ — Digital

Digital (53, 82, 2969)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸², 2969, F₁₆, 29) (dual of [2969, 2887, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸², 4096, F₁₆, 29) (dual of [4096, 4014, 30]-code), using
  - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(53, 53+29, 3733015)-Net in Base 16 — Upper bound on s

There is no (53, 82, 3733016)-net in base 16, because

1 times m-reduction [i] would yield (53, 81, 3733016)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 175848 089262 353158 962565 962569 333020 047874 180875 058351 155407 318284 675899 941992 801647 607839 562086 > 16⁸¹ [i]