MinT - Best Known (207, 225, s)-Nets in Base 2

Best Known (207, 225, s)-Nets in Base 2

(207, 225, 932087)-Net over F₂ — Constructive and digital

Digital (207, 225, 932087)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 18, 20)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (189, 207, 932067)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁰⁷, 932067, F₂, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(2²⁰⁷, large, F₂, 18) (dual of [large, large−207, 19]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(207, 225, 1677740)-Net over F₂ — Digital

Digital (207, 225, 1677740)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²²⁵, 1677740, F₂, 5, 18) (dual of [(1677740, 5), 8388475, 19]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2¹⁸, 20, F₂, 5, 9) (dual of [(20, 5), 82, 10]-NRT-code), using
    - extracting embedded OOA [i] based on digital (9, 18, 20)-net over F₂, using
      - linear code embedding found in a computer search [i]
  2. linear OOA(2²⁰⁷, 1677720, F₂, 5, 18) (dual of [(1677720, 5), 8388393, 19]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2²⁰⁷, 8388600, F₂, 18) (dual of [8388600, 8388393, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²⁰⁷, large, F₂, 18) (dual of [large, large−207, 19]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(207, 225, large)-Net in Base 2 — Upper bound on s

There is no (207, 225, large)-net in base 2, because

16 times m-reduction [i] would yield (207, 209, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]