MinT - Best Known (139−116, 139, s)-Nets in Base 3

Best Known (139−116, 139, s)-Nets in Base 3

(139−116, 139, 32)-Net over F₃ — Constructive and digital

Digital (23, 139, 32)-net over F₃, using

t-expansion [i] based on digital (21, 139, 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]

(139−116, 139, 61)-Net in Base 3 — Upper bound on s

There is no (23, 139, 62)-net in base 3, because

21 times m-reduction [i] would yield (23, 118, 62)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹¹⁸, 62, S₃, 2, 95), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 6589 037766 352914 438181 659374 868551 812348 743967 638787 390137 / 32 > 3¹¹⁸ [i]