MinT - Best Known (234−41, 234, s)-Nets in Base 2

Best Known (234−41, 234, s)-Nets in Base 2

(234−41, 234, 380)-Net over F₂ — Constructive and digital

Digital (193, 234, 380)-net over F₂, using

1 times m-reduction [i] based on digital (193, 235, 380)-net over F₂, using
- trace code for nets [i] based on digital (5, 47, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(234−41, 234, 980)-Net over F₂ — Digital

Digital (193, 234, 980)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁴, 980, F₂, 2, 41) (dual of [(980, 2), 1726, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³⁴, 1047, F₂, 2, 41) (dual of [(1047, 2), 1860, 42]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²³⁴, 2094, F₂, 41) (dual of [2094, 1860, 42]-code), using
    - constructionÂ X applied to C_e(40) âŠ‚ C_e(34) [i] based on
      1. linear OA(2²²¹, 2048, F₂, 41) (dual of [2048, 1827, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      2. linear OA(2¹⁸⁸, 2048, F₂, 35) (dual of [2048, 1860, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      3. linear OA(2¹³, 46, F₂, 5) (dual of [46, 33, 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]

(234−41, 234, 26657)-Net in Base 2 — Upper bound on s

There is no (193, 234, 26658)-net in base 2, because

1 times m-reduction [i] would yield (193, 233, 26658)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13804 745247 728502 178171 524668 273891 148469 194096 344570 715854 331757 836808 > 2²³³ [i]