MinT - Best Known (84, 130, s)-Nets in Base 16

Best Known (84, 130, s)-Nets in Base 16

(84, 130, 587)-Net over F₁₆ — Constructive and digital

Digital (84, 130, 587)-net over F₁₆, using

16¹ times duplication [i] based on digital (83, 129, 587)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 29, 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 (54, 100, 522)-net over F₁₆, using
    - trace code for nets [i] based on digital (4, 50, 261)-net over F₂₅₆, using
      - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(84, 130, 612)-Net in Base 16 — Constructive

(84, 130, 612)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (15, 38, 98)-net in base 16, using
  - 2 times m-reduction [i] based on (15, 40, 98)-net in base 16, using
    - base change [i] based on digital (7, 32, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]
2. digital (46, 92, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 46, 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]

(84, 130, 3877)-Net over F₁₆ — Digital

Digital (84, 130, 3877)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹³⁰, 3877, F₁₆, 46) (dual of [3877, 3747, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(16¹³⁰, 4096, F₁₆, 46) (dual of [4096, 3966, 47]-code), using
  - an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]

(84, 130, 4020307)-Net in Base 16 — Upper bound on s

There is no (84, 130, 4020308)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 432417 997044 888747 558120 090231 673183 758108 776672 924577 557773 672974 580935 568651 543237 196399 209382 720657 836243 144928 281073 011565 326590 600968 052645 484986 709936 > 16¹³⁰ [i]