MinT - Best Known (216, 216+18, s)-Nets in Base 2

Best Known (216, 216+18, s)-Nets in Base 2

(216, 216+18, 932107)-Net over F₂ — Constructive and digital

Digital (216, 234, 932107)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (18, 27, 40)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁷, 40, F₂, 9, 9) (dual of [(40, 9), 333, 10]-NRT-code), using
    - appending kth column [i] based on linear OOA(2²⁷, 40, F₂, 8, 9) (dual of [(40, 8), 293, 10]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2⁹, 23, F₂, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
        appending 4 arbitrary columns [i] based on linear OOA(2⁹, 23, F₂, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [i]
        linear OOA(2¹⁸, 20, F₂, 8, 9) (dual of [(20, 8), 142, 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. 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]

(216, 216+18, 1677760)-Net over F₂ — Digital

Digital (216, 234, 1677760)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁴, 1677760, F₂, 5, 18) (dual of [(1677760, 5), 8388566, 19]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁷, 40, F₂, 5, 9) (dual of [(40, 5), 173, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁹, 23, F₂, 5, 4) (dual of [(23, 5), 106, 5]-NRT-code), using
        appending 1 arbitrary column [i] based on linear OOA(2⁹, 23, F₂, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [i]
      2. 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]

(216, 216+18, large)-Net in Base 2 — Upper bound on s

There is no (216, 234, large)-net in base 2, because

16 times m-reduction [i] would yield (216, 218, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (216, 216+18, s)-Nets in Base 2

(216, 216+18, 932107)-Net over F2 — Constructive and digital

(216, 216+18, 1677760)-Net over F2 — Digital

(216, 216+18, large)-Net in Base 2 — Upper bound on s

(216, 216+18, 932107)-Net over F₂ — Constructive and digital

(216, 216+18, 1677760)-Net over F₂ — Digital