MinT - Best Known (150, 150+13, s)-Nets in Base 2

Best Known (150, 150+13, s)-Nets in Base 2

(150, 150+13, 1398230)-Net over F₂ — Constructive and digital

Digital (150, 163, 1398230)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (18, 24, 130)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁴, 130, F₂, 6, 6) (dual of [(130, 6), 756, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(2²⁴, 130, F₂, 5, 6) (dual of [(130, 5), 626, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2³, 3, F₂, 5, 3) (dual of [(3, 5), 12, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(5;12,2) [i]
        linear OOA(2²¹, 127, F₂, 5, 6) (dual of [(127, 5), 614, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (15, 21, 127)-net over F₂, using
        linear code embedding found in a computer search [i]
2. digital (126, 139, 1398100)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹³⁹, 1398100, F₂, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(2¹³⁹, 8388601, F₂, 13) (dual of [8388601, 8388462, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(150, 150+13, 2097280)-Net over F₂ — Digital

Digital (150, 163, 2097280)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶³, 2097280, F₂, 4, 13) (dual of [(2097280, 4), 8388957, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁴, 130, F₂, 4, 6) (dual of [(130, 4), 496, 7]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2³, 3, F₂, 4, 3) (dual of [(3, 4), 9, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;9,2) [i]
      2. linear OOA(2²¹, 127, F₂, 4, 6) (dual of [(127, 4), 487, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (15, 21, 127)-net over F₂, using
        linear code embedding found in a computer search [i]
  2. linear OOA(2¹³⁹, 2097150, F₂, 4, 13) (dual of [(2097150, 4), 8388461, 14]-NRT-code), using
    - OOA 4-folding [i] based on linear OA(2¹³⁹, 8388600, F₂, 13) (dual of [8388600, 8388461, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(150, 150+13, large)-Net in Base 2 — Upper bound on s

There is no (150, 163, large)-net in base 2, because

11 times m-reduction [i] would yield (150, 152, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]