MinT - Best Known (238−41, 238, s)-Nets in Base 2

Best Known (238−41, 238, s)-Nets in Base 2

(238−41, 238, 380)-Net over F₂ — Constructive and digital

Digital (197, 238, 380)-net over F₂, using

2 times m-reduction [i] based on digital (197, 240, 380)-net over F₂, using
- trace code for nets [i] based on digital (5, 48, 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]

(238−41, 238, 1049)-Net over F₂ — Digital

Digital (197, 238, 1049)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁸, 1049, F₂, 2, 41) (dual of [(1049, 2), 1860, 42]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] 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]

(238−41, 238, 30626)-Net in Base 2 — Upper bound on s

There is no (197, 238, 30627)-net in base 2, because

1 times m-reduction [i] would yield (197, 237, 30627)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 220964 819293 276843 555068 176311 348977 386994 453140 511587 148159 205280 215336 > 2²³⁷ [i]