MinT - Best Known (245−21, 245, s)-Nets in Base 2

Best Known (245−21, 245, s)-Nets in Base 2

(245−21, 245, 838868)-Net over F₂ — Constructive and digital

Digital (224, 245, 838868)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 8)-net over F₂, using
  - net from sequence [i] based on digital (4, 7)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 4 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (210, 231, 838860)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²³¹, 838860, F₂, 21, 21) (dual of [(838860, 21), 17615829, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(2²³¹, 8388601, F₂, 21) (dual of [8388601, 8388370, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²³¹, large, F₂, 21) (dual of [large, large−231, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(245−21, 245, 1198379)-Net over F₂ — Digital

Digital (224, 245, 1198379)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁵, 1198379, F₂, 7, 21) (dual of [(1198379, 7), 8388408, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2¹⁴, 8, F₂, 7, 10) (dual of [(8, 7), 42, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(7;F,45P) [i] based on function field F/F₂ with g(F) = 4 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(2²³¹, 1198371, F₂, 7, 21) (dual of [(1198371, 7), 8388366, 22]-NRT-code), using
    - OOA 7-folding [i] based on linear OA(2²³¹, 8388597, F₂, 21) (dual of [8388597, 8388366, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²³¹, large, F₂, 21) (dual of [large, large−231, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(245−21, 245, large)-Net in Base 2 — Upper bound on s

There is no (224, 245, large)-net in base 2, because

19 times m-reduction [i] would yield (224, 226, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]