MinT - Best Known (235−35, 235, s)-Nets in Base 2

Best Known (235−35, 235, s)-Nets in Base 2

(235−35, 235, 624)-Net over F₂ — Constructive and digital

Digital (200, 235, 624)-net over F₂, using

2¹ times duplication [i] based on digital (199, 234, 624)-net over F₂, using
- t-expansion [i] based on digital (198, 234, 624)-net over F₂, using
  - trace code for nets [i] based on digital (3, 39, 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]

(235−35, 235, 2178)-Net over F₂ — Digital

Digital (200, 235, 2178)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁵, 2178, F₂, 3, 35) (dual of [(2178, 3), 6299, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³⁵, 2748, F₂, 3, 35) (dual of [(2748, 3), 8009, 36]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2²³⁵, 8244, F₂, 35) (dual of [8244, 8009, 36]-code), using
    - constructionÂ X applied to C_e(34) âŠ‚ C_e(28) [i] based on
      1. linear OA(2²²², 8192, F₂, 35) (dual of [8192, 7970, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      2. linear OA(2¹⁸³, 8192, F₂, 29) (dual of [8192, 8009, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(2¹³, 52, F₂, 5) (dual of [52, 39, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹³, 63, F₂, 5) (dual of [63, 50, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 2⁶âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(235−35, 235, 99867)-Net in Base 2 — Upper bound on s

There is no (200, 235, 99868)-net in base 2, because

1 times m-reduction [i] would yield (200, 234, 99868)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27607 509399 413947 873296 114512 057730 968864 998994 751252 250586 636180 767499 > 2²³⁴ [i]