MinT - Best Known (210−34, 210, s)-Nets in Base 2

Best Known (210−34, 210, s)-Nets in Base 2

(210−34, 210, 490)-Net over F₂ — Constructive and digital

Digital (176, 210, 490)-net over F₂, using

t-expansion [i] based on digital (175, 210, 490)-net over F₂, using
- trace code for nets [i] based on digital (7, 42, 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]

(210−34, 210, 1375)-Net over F₂ — Digital

Digital (176, 210, 1375)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹⁰, 1375, F₂, 3, 34) (dual of [(1375, 3), 3915, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2²¹⁰, 4125, F₂, 34) (dual of [4125, 3915, 35]-code), using
  - 1 times truncation [i] based on linear OA(2²¹¹, 4126, F₂, 35) (dual of [4126, 3915, 36]-code), using
    - constructionÂ X applied to C_e(34) âŠ‚ C_e(30) [i] based on
      1. 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]
      2. linear OA(2¹⁸¹, 4096, F₂, 31) (dual of [4096, 3915, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      3. 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]

(210−34, 210, 37519)-Net in Base 2 — Upper bound on s

There is no (176, 210, 37520)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1645 568044 812917 175173 017888 385769 209958 652895 606184 240889 472328 > 2²¹⁰ [i]