MinT - Best Known (259−20, 259, s)-Nets in Base 2

Best Known (259−20, 259, s)-Nets in Base 2

(259−20, 259, 838895)-Net over F₂ — Constructive and digital

Digital (239, 259, 838895)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 29, 35)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (4, 9, 14)-net over F₂, using
      - linear code embedding found in a computer search [i]
    2. digital (10, 20, 21)-net over F₂, using
      - linear code embedding found in a computer search [i]
2. digital (210, 230, 838860)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²³⁰, 838860, F₂, 20, 20) (dual of [(838860, 20), 16776970, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(2²³⁰, 8388600, F₂, 20) (dual of [8388600, 8388370, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²³⁰, large, F₂, 20) (dual of [large, large−230, 21]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(259−20, 259, 1677755)-Net over F₂ — Digital

Digital (239, 259, 1677755)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁵⁹, 1677755, F₂, 5, 20) (dual of [(1677755, 5), 8388516, 21]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁹, 35, F₂, 5, 10) (dual of [(35, 5), 146, 11]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁹, 14, F₂, 5, 5) (dual of [(14, 5), 61, 6]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 14, F₂, 4, 5) (dual of [(14, 4), 47, 6]-NRT-code), using
        extracting embedded OOA [i] based on digital (4, 9, 14)-net over F₂, using
        linear code embedding found in a computer search [i]
      2. linear OOA(2²⁰, 21, F₂, 5, 10) (dual of [(21, 5), 85, 11]-NRT-code), using
        extracting embedded OOA [i] based on digital (10, 20, 21)-net over F₂, using
        linear code embedding found in a computer search [i]
  2. linear OOA(2²³⁰, 1677720, F₂, 5, 20) (dual of [(1677720, 5), 8388370, 21]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2²³⁰, 8388600, F₂, 20) (dual of [8388600, 8388370, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²³⁰, large, F₂, 20) (dual of [large, large−230, 21]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(259−20, 259, large)-Net in Base 2 — Upper bound on s

There is no (239, 259, large)-net in base 2, because

18 times m-reduction [i] would yield (239, 241, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]