MinT - Best Known (70−25, 70, s)-Nets in Base 16

Best Known (70−25, 70, s)-Nets in Base 16

(70−25, 70, 581)-Net over F₁₆ — Constructive and digital

Digital (45, 70, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 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 (27, 52, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 26, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(70−25, 70, 594)-Net in Base 16 — Constructive

(45, 70, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (8, 20, 80)-net in base 16, using
  - 1 times m-reduction [i] based on (8, 21, 80)-net in base 16, using
    - base change [i] based on digital (1, 14, 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 (25, 50, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 25, 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]

(70−25, 70, 2563)-Net over F₁₆ — Digital

Digital (45, 70, 2563)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁰, 2563, F₁₆, 25) (dual of [2563, 2493, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁷⁰, 4096, F₁₆, 25) (dual of [4096, 4026, 26]-code), using
  - an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(70−25, 70, 2957734)-Net in Base 16 — Upper bound on s

There is no (45, 70, 2957735)-net in base 16, because

1 times m-reduction [i] would yield (45, 69, 2957735)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 121417 244230 523690 617023 247789 026182 089651 110014 643835 924008 043545 855670 459378 179676 > 16⁶⁹ [i]