MinT - Best Known (28−9, 28, s)-Nets in Base 2

Best Known (28−9, 28, s)-Nets in Base 2

(28−9, 28, 46)-Net over F₂ — Constructive and digital

Digital (19, 28, 46)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁸, 46, F₂, 9, 9) (dual of [(46, 9), 386, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2²⁸, 46, F₂, 8, 9) (dual of [(46, 8), 340, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. linear OOA(2¹⁹, 23, F₂, 8, 9) (dual of [(23, 8), 165, 10]-NRT-code), using
      - extracting embedded OOA [i] based on digital (10, 19, 23)-net over F₂, using
        linear code embedding found in a computer search [i]

(28−9, 28, 47)-Net over F₂ — Digital

Digital (19, 28, 47)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁸, 47, F₂, 9, 9) (dual of [(47, 9), 395, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2²⁸, 47, F₂, 8, 9) (dual of [(47, 8), 348, 10]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁸, 47, F₂, 2, 9) (dual of [(47, 2), 66, 10]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2²⁸, 94, F₂, 9) (dual of [94, 66, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²⁸, 95, F₂, 9) (dual of [95, 67, 10]-code), using
        adding a parity check bit [i] based on linear OA(2²⁷, 94, F₂, 8) (dual of [94, 67, 9]-code), using
        a â€œGraâ€ code from Grasslâ€™s database [i]

(28−9, 28, 232)-Net in Base 2 — Upper bound on s

There is no (19, 28, 233)-net in base 2, because

1 times m-reduction [i] would yield (19, 27, 233)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 134 697534 > 2²⁷ [i]