MinT - Best Known (235−39, 235, s)-Nets in Base 2

Best Known (235−39, 235, s)-Nets in Base 2

(235−39, 235, 490)-Net over F₂ — Constructive and digital

Digital (196, 235, 490)-net over F₂, using

t-expansion [i] based on digital (195, 235, 490)-net over F₂, using
- trace code for nets [i] based on digital (7, 47, 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]

(235−39, 235, 1326)-Net over F₂ — Digital

Digital (196, 235, 1326)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁵, 1326, F₂, 3, 39) (dual of [(1326, 3), 3743, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³⁵, 1375, F₂, 3, 39) (dual of [(1375, 3), 3890, 40]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2²³⁵, 4125, F₂, 39) (dual of [4125, 3890, 40]-code), using
    - discarding factors / shortening the dual code based on linear OA(2²³⁵, 4126, F₂, 39) (dual of [4126, 3891, 40]-code), using
      - constructionÂ X applied to C_e(38) âŠ‚ C_e(34) [i] based on
        linear OA(2²²⁹, 4096, F₂, 39) (dual of [4096, 3867, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
        linear OA(2²⁰⁵, 4096, F₂, 35) (dual of [4096, 3891, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
        linear OA(2⁶, 30, F₂, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
        discarding factors / shortening the dual code based on 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]

(235−39, 235, 40395)-Net in Base 2 — Upper bound on s

There is no (196, 235, 40396)-net in base 2, because

1 times m-reduction [i] would yield (196, 234, 40396)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27608 956734 250311 401088 003728 838551 554396 637901 949757 297177 866344 587396 > 2²³⁴ [i]