MinT - Best Known (213−191, 213, s)-Nets in Base 3

Best Known (213−191, 213, s)-Nets in Base 3

(213−191, 213, 32)-Net over F₃ — Constructive and digital

Digital (22, 213, 32)-net over F₃, using

t-expansion [i] based on digital (21, 213, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(213−191, 213, 55)-Net in Base 3 — Upper bound on s

There is no (22, 213, 56)-net in base 3, because

51 times m-reduction [i] would yield (22, 162, 56)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶², 56, S₃, 3, 140), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 9 591030 573196 414342 037258 204812 713227 586934 996056 698075 676829 713463 027303 933239 / 47 > 3¹⁶² [i]