MinT - Best Known (253−197, 253, s)-Nets in Base 2

Best Known (253−197, 253, s)-Nets in Base 2

(253−197, 253, 42)-Net over F₂ — Constructive and digital

Digital (56, 253, 42)-net over F₂, using

t-expansion [i] based on digital (54, 253, 42)-net over F₂, using
- net from sequence [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(253−197, 253, 72)-Net in Base 2 — Upper bound on s

There is no (56, 253, 73)-net in base 2, because

43 times m-reduction [i] would yield (56, 210, 73)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²¹⁰, 73, S₂, 3, 154), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 842498 333348 457493 583344 221469 363458 551160 763204 392890 034487 820288 / 465 > 2²¹⁰ [i]