MinT - Best Known (192−29, 192, s)-Nets in Base 2

Best Known (192−29, 192, s)-Nets in Base 2

(192−29, 192, 624)-Net over F₂ — Constructive and digital

Digital (163, 192, 624)-net over F₂, using

trace code for nets [i] based on digital (3, 32, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(192−29, 192, 2057)-Net over F₂ — Digital

Digital (163, 192, 2057)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹², 2057, F₂, 4, 29) (dual of [(2057, 4), 8036, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2¹⁹², 8228, F₂, 29) (dual of [8228, 8036, 30]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(2¹⁸⁹, 8225, F₂, 29) (dual of [8225, 8036, 30]-code), using
    - constructionÂ X applied to C([0,14]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(2¹⁸³, 8193, F₂, 29) (dual of [8193, 8010, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
      2. linear OA(2¹⁵⁷, 8193, F₂, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. 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]

(192−29, 192, 77312)-Net in Base 2 — Upper bound on s

There is no (163, 192, 77313)-net in base 2, because

1 times m-reduction [i] would yield (163, 191, 77313)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3138 620310 186759 927754 761169 058200 922262 377596 604416 710912 > 2¹⁹¹ [i]