MinT - Best Known (169−33, 169, s)-Nets in Base 4

Best Known (169−33, 169, s)-Nets in Base 4

(169−33, 169, 1094)-Net over F₄ — Constructive and digital

Digital (136, 169, 1094)-net over F₄, using

4¹ times duplication [i] based on digital (135, 168, 1094)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 36, 66)-net over F₄, using
    - trace code for nets [i] based on digital (2, 18, 33)-net over F₁₆, using
      - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
  2. digital (99, 132, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 33, 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]

(169−33, 169, 8192)-Net over F₄ — Digital

Digital (136, 169, 8192)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁶⁹, 8192, F₄, 2, 33) (dual of [(8192, 2), 16215, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4¹⁶⁹, 16384, F₄, 33) (dual of [16384, 16215, 34]-code), using
  - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(169−33, 169, 4753858)-Net in Base 4 — Upper bound on s

There is no (136, 169, 4753859)-net in base 4, because

1 times m-reduction [i] would yield (136, 168, 4753859)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 139984 510153 021819 550672 230185 806966 589676 299856 029467 578344 202701 168000 889696 789657 484746 895244 323044 > 4¹⁶⁸ [i]