MinT - Best Known (29, 29+10, s)-Nets in Base 2

Best Known (29, 29+10, s)-Nets in Base 2

(29, 29+10, 72)-Net over F₂ — Constructive and digital

Digital (29, 39, 72)-net over F₂, using

trace code for nets [i] based on digital (3, 13, 24)-net over F₈, using
- net from sequence [i] based on digital (3, 23)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
    - the Klein quartic over F₈ [i]

(29, 29+10, 95)-Net over F₂ — Digital

Digital (29, 39, 95)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³⁹, 95, F₂, 2, 10) (dual of [(95, 2), 151, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2³⁹, 190, F₂, 10) (dual of [190, 151, 11]-code), using
  - discarding factors / shortening the dual code based on linear OA(2³⁹, 191, F₂, 10) (dual of [191, 152, 11]-code), using
    - a â€œXâ€ code from Brouwerâ€™s database [i]

(29, 29+10, 573)-Net in Base 2 — Upper bound on s

There is no (29, 39, 574)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 551524 708561 > 2³⁹ [i]