MinT - Best Known (215−33, 215, s)-Nets in Base 2

Best Known (215−33, 215, s)-Nets in Base 2

(215−33, 215, 520)-Net over F₂ — Constructive and digital

Digital (182, 215, 520)-net over F₂, using

t-expansion [i] based on digital (181, 215, 520)-net over F₂, using
- trace code for nets [i] based on digital (9, 43, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(215−33, 215, 2056)-Net over F₂ — Digital

Digital (182, 215, 2056)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹⁵, 2056, F₂, 4, 33) (dual of [(2056, 4), 8009, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2²¹⁵, 8224, F₂, 33) (dual of [8224, 8009, 34]-code), using
  - discarding factors / shortening the dual code based on linear OA(2²¹⁵, 8225, F₂, 33) (dual of [8225, 8010, 34]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
      1. linear OA(2²⁰⁹, 8193, F₂, 33) (dual of [8193, 7984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. linear OA(2¹⁸³, 8193, F₂, 29) (dual of [8193, 8010, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
      3. linear OA(2⁶, 32, F₂, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
        Reedâ€“Muller code RM(1,5) [i]
        caps in base bÂ =Â 2 [i]

(215−33, 215, 72222)-Net in Base 2 — Upper bound on s

There is no (182, 215, 72223)-net in base 2, because

1 times m-reduction [i] would yield (182, 214, 72223)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 26329 891625 405029 717071 801865 526864 154942 144327 047646 616247 074259 > 2²¹⁴ [i]