MinT - Best Known (248−41, 248, s)-Nets in Base 2

Best Known (248−41, 248, s)-Nets in Base 2

(248−41, 248, 490)-Net over F₂ — Constructive and digital

Digital (207, 248, 490)-net over F₂, using

2 times m-reduction [i] based on digital (207, 250, 490)-net over F₂, using
- trace code for nets [i] based on digital (7, 50, 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]

(248−41, 248, 1376)-Net over F₂ — Digital

Digital (207, 248, 1376)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁸, 1376, F₂, 3, 41) (dual of [(1376, 3), 3880, 42]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2²⁴⁸, 4128, F₂, 41) (dual of [4128, 3880, 42]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(2²⁴⁷, 4127, F₂, 41) (dual of [4127, 3880, 42]-code), using
    - constructionÂ X applied to C([0,20]) âŠ‚ C([0,18]) [i] based on
      1. linear OA(2²⁴¹, 4097, F₂, 41) (dual of [4097, 3856, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 2²⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
      2. linear OA(2²¹⁷, 4097, F₂, 37) (dual of [4097, 3880, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 2²⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [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]

(248−41, 248, 43324)-Net in Base 2 — Upper bound on s

There is no (207, 248, 43325)-net in base 2, because

1 times m-reduction [i] would yield (207, 247, 43325)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 226 233643 503267 708909 894028 744465 492447 524697 507424 744541 217544 608290 683911 > 2²⁴⁷ [i]