MinT - Best Known (144, 144+33, s)-Nets in Base 2

Best Known (144, 144+33, s)-Nets in Base 2

(144, 144+33, 269)-Net over F₂ — Constructive and digital

Digital (144, 177, 269)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (123, 156, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 39, 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]

(144, 144+33, 683)-Net over F₂ — Digital

Digital (144, 177, 683)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷⁷, 683, F₂, 3, 33) (dual of [(683, 3), 1872, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2¹⁷⁷, 2049, F₂, 33) (dual of [2049, 1872, 34]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(144, 144+33, 13903)-Net in Base 2 — Upper bound on s

There is no (144, 177, 13904)-net in base 2, because

1 times m-reduction [i] would yield (144, 176, 13904)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 95794 838785 060387 464386 717689 441061 447822 299199 613060 > 2¹⁷⁶ [i]