MinT - Best Known (62−28, 62, s)-Nets in Base 2

Best Known (62−28, 62, s)-Nets in Base 2

(62−28, 62, 28)-Net over F₂ — Constructive and digital

Digital (34, 62, 28)-net over F₂, using

trace code for nets [i] based on digital (3, 31, 14)-net over F₄, using
- net from sequence [i] based on digital (3, 13)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
    - an explicitly constructive algebraic function field [i]

(62−28, 62, 29)-Net over F₂ — Digital

Digital (34, 62, 29)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−28, 62, 106)-Net over F₂ — Upper bound on s (digital)

There is no digital (34, 62, 107)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁶², 107, F₂, 28) (dual of [107, 45, 29]-code), but
- adding a parity check bit [i] would yield linear OA(2⁶³, 108, F₂, 29) (dual of [108, 45, 30]-code), but
  - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(62−28, 62, 107)-Net in Base 2 — Upper bound on s

There is no (34, 62, 108)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁶², 108, S₂, 28), but
- the linear programming bound shows that MÂ â‰¥ 2852 194618 252966 848277 066929 703881 801728 / 562 984193 820668 484375 > 2⁶² [i]