MinT - Best Known (216, 216+19, s)-Nets in Base 2

Best Known (216, 216+19, s)-Nets in Base 2

(216, 216+19, 932106)-Net over F₂ — Constructive and digital

Digital (216, 235, 932106)-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, 208, 932066)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁰⁸, 932066, F₂, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(2²⁰⁸, 8388595, F₂, 19) (dual of [8388595, 8388387, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²⁰⁸, large, F₂, 19) (dual of [large, large−208, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(216, 216+19, 1398140)-Net over F₂ — Digital

Digital (216, 235, 1398140)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁵, 1398140, F₂, 6, 19) (dual of [(1398140, 6), 8388605, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁷, 40, F₂, 6, 9) (dual of [(40, 6), 213, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁹, 23, F₂, 6, 4) (dual of [(23, 6), 129, 5]-NRT-code), using
        appending 2 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]
      2. linear OOA(2¹⁸, 20, F₂, 6, 9) (dual of [(20, 6), 102, 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(2²⁰⁸, 1398100, F₂, 6, 19) (dual of [(1398100, 6), 8388392, 20]-NRT-code), using
    - OOA 6-folding [i] based on linear OA(2²⁰⁸, 8388600, F₂, 19) (dual of [8388600, 8388392, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²⁰⁸, large, F₂, 19) (dual of [large, large−208, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(216, 216+19, large)-Net in Base 2 — Upper bound on s

There is no (216, 235, large)-net in base 2, because

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

Best Known (216, 216+19, s)-Nets in Base 2

(216, 216+19, 932106)-Net over F2 — Constructive and digital

(216, 216+19, 1398140)-Net over F2 — Digital

(216, 216+19, large)-Net in Base 2 — Upper bound on s

(216, 216+19, 932106)-Net over F₂ — Constructive and digital

(216, 216+19, 1398140)-Net over F₂ — Digital