MinT - Best Known (241−37, 241, s)-Nets in Base 2

Best Known (241−37, 241, s)-Nets in Base 2

(241−37, 241, 624)-Net over F₂ — Constructive and digital

Digital (204, 241, 624)-net over F₂, using

2¹ times duplication [i] based on digital (203, 240, 624)-net over F₂, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(241−37, 241, 2056)-Net over F₂ — Digital

Digital (204, 241, 2056)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴¹, 2056, F₂, 4, 37) (dual of [(2056, 4), 7983, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2²⁴¹, 8224, F₂, 37) (dual of [8224, 7983, 38]-code), using
  - discarding factors / shortening the dual code based on linear OA(2²⁴¹, 8225, F₂, 37) (dual of [8225, 7984, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(2²³⁵, 8193, F₂, 37) (dual of [8193, 7958, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      2. 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]
      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]

(241−37, 241, 77931)-Net in Base 2 — Upper bound on s

There is no (204, 241, 77932)-net in base 2, because

1 times m-reduction [i] would yield (204, 240, 77932)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 766855 796592 557127 114107 060669 354388 497878 147033 582287 935365 379922 591130 > 2²⁴⁰ [i]