MinT - Best Known (216−36, 216, s)-Nets in Base 2

Best Known (216−36, 216, s)-Nets in Base 2

(216−36, 216, 490)-Net over F₂ — Constructive and digital

Digital (180, 216, 490)-net over F₂, using

2¹ times duplication [i] based on digital (179, 215, 490)-net over F₂, using
- trace code for nets [i] based on digital (7, 43, 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]

(216−36, 216, 1244)-Net over F₂ — Digital

Digital (180, 216, 1244)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹⁶, 1244, F₂, 3, 36) (dual of [(1244, 3), 3516, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²¹⁶, 1365, F₂, 3, 36) (dual of [(1365, 3), 3879, 37]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2²¹⁶, 4095, F₂, 36) (dual of [4095, 3879, 37]-code), using
    - discarding factors / shortening the dual code based on linear OA(2²¹⁶, 4096, F₂, 36) (dual of [4096, 3880, 37]-code), using
      - 1 times truncation [i] based on linear OA(2²¹⁷, 4097, F₂, 37) (dual of [4097, 3880, 38]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4097Â | 2²⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]

(216−36, 216, 30911)-Net in Base 2 — Upper bound on s

There is no (180, 216, 30912)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 105352 038017 248692 205640 998139 341016 337841 078312 755520 208231 904925 > 2²¹⁶ [i]