MinT - Best Known (198, 237, s)-Nets in Base 2

Best Known (198, 237, s)-Nets in Base 2

(198, 237, 490)-Net over F₂ — Constructive and digital

Digital (198, 237, 490)-net over F₂, using

2² times duplication [i] based on 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]

(198, 237, 1376)-Net over F₂ — Digital

Digital (198, 237, 1376)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁷, 1376, F₂, 3, 39) (dual of [(1376, 3), 3891, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2²³⁷, 4128, F₂, 39) (dual of [4128, 3891, 40]-code), using
  - 2 times code embedding in larger space [i] 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
      1. 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]
      2. 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]
      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]

(198, 237, 43455)-Net in Base 2 — Upper bound on s

There is no (198, 237, 43456)-net in base 2, because

1 times m-reduction [i] would yield (198, 236, 43456)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 110449 581790 803559 816979 445757 757337 321180 218294 658751 348022 518906 687405 > 2²³⁶ [i]