MinT - Best Known (46−17, 46, s)-Nets in Base 16

Best Known (46−17, 46, s)-Nets in Base 16

(46−17, 46, 559)-Net over F₁₆ — Constructive and digital

Digital (29, 46, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (17, 34, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 17, 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]

(46−17, 46, 579)-Net in Base 16 — Constructive

(29, 46, 579)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (4, 12, 65)-net in base 16, using
  - base change [i] based on digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (17, 34, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 17, 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]

(46−17, 46, 2048)-Net over F₁₆ — Digital

Digital (29, 46, 2048)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁶, 2048, F₁₆, 2, 17) (dual of [(2048, 2), 4050, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(16⁴⁶, 4096, F₁₆, 17) (dual of [4096, 4050, 18]-code), using
  - an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(46−17, 46, 1488581)-Net in Base 16 — Upper bound on s

There is no (29, 46, 1488582)-net in base 16, because

1 times m-reduction [i] would yield (29, 45, 1488582)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 532502 064463 793269 068791 225546 967611 214873 232381 466616 > 16⁴⁵ [i]