MinT - Best Known (66, 103, s)-Nets in Base 16

Best Known (66, 103, s)-Nets in Base 16

(66, 103, 583)-Net over F₁₆ — Constructive and digital

Digital (66, 103, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (65, 102, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 24, 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 (41, 78, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 39, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(66, 103, 594)-Net in Base 16 — Constructive

(66, 103, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (11, 29, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (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 (37, 74, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 37, 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]

(66, 103, 2976)-Net over F₁₆ — Digital

Digital (66, 103, 2976)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰³, 2976, F₁₆, 37) (dual of [2976, 2873, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹⁰³, 4096, F₁₆, 37) (dual of [4096, 3993, 38]-code), using
  - an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]

(66, 103, 3352618)-Net in Base 16 — Upper bound on s

There is no (66, 103, 3352619)-net in base 16, because

1 times m-reduction [i] would yield (66, 102, 3352619)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 661 058501 460272 382282 767340 707042 478420 703046 665767 972727 074758 679001 287916 441274 944785 365349 488266 548479 953166 469796 268506 > 16¹⁰² [i]