MinT - Best Known (47, 47+15, s)-Nets in Base 2

Best Known (47, 47+15, s)-Nets in Base 2

(47, 47+15, 90)-Net over F₂ — Constructive and digital

Digital (47, 62, 90)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (12, 19, 47)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (28, 43, 45)-net over F₂, using
  - 2 times m-reduction [i] based on digital (28, 45, 45)-net over F₂, using
    - a shift-net [i]

(47, 47+15, 139)-Net over F₂ — Digital

Digital (47, 62, 139)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶², 139, F₂, 2, 15) (dual of [(139, 2), 216, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁶², 278, F₂, 15) (dual of [278, 216, 16]-code), using
  - extended or lengthened BCH codes [i]

(47, 47+15, 1409)-Net in Base 2 — Upper bound on s

There is no (47, 62, 1410)-net in base 2, because

1 times m-reduction [i] would yield (47, 61, 1410)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 309520 154234 686848 > 2⁶¹ [i]