MinT - Best Known (238−43, 238, s)-Nets in Base 2

Best Known (238−43, 238, s)-Nets in Base 2

(238−43, 238, 320)-Net over F₂ — Constructive and digital

Digital (195, 238, 320)-net over F₂, using

2 times m-reduction [i] based on digital (195, 240, 320)-net over F₂, using
- trace code for nets [i] based on digital (3, 48, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]

(238−43, 238, 885)-Net over F₂ — Digital

Digital (195, 238, 885)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁸, 885, F₂, 2, 43) (dual of [(885, 2), 1532, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³⁸, 1038, F₂, 2, 43) (dual of [(1038, 2), 1838, 44]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²³⁸, 2076, F₂, 43) (dual of [2076, 1838, 44]-code), using
    - constructionÂ X applied to C_e(42) âŠ‚ C_e(38) [i] based on
      1. linear OA(2²³², 2048, F₂, 43) (dual of [2048, 1816, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
      2. linear OA(2²¹⁰, 2048, F₂, 39) (dual of [2048, 1838, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
      3. linear OA(2⁶, 28, F₂, 3) (dual of [28, 22, 4]-code or 28-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]

(238−43, 238, 21637)-Net in Base 2 — Upper bound on s

There is no (195, 238, 21638)-net in base 2, because

1 times m-reduction [i] would yield (195, 237, 21638)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 220934 654851 432200 849186 091060 478576 772924 645872 101976 863258 514465 311816 > 2²³⁷ [i]