MinT - Best Known (140−127, 140, s)-Nets in Base 4

Best Known (140−127, 140, s)-Nets in Base 4

(140−127, 140, 30)-Net over F₄ — Constructive and digital

Digital (13, 140, 30)-net over F₄, using

net from sequence [i] based on digital (13, 29)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
  - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(140−127, 140, 33)-Net over F₄ — Digital

Digital (13, 140, 33)-net over F₄, using

net from sequence [i] based on digital (13, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
  - Drinfeld modules of rank 1 [i]

(140−127, 140, 52)-Net in Base 4 — Upper bound on s

There is no (13, 140, 53)-net in base 4, because

39 times m-reduction [i] would yield (13, 101, 53)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁰¹, 53, S₄, 2, 88), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 668 486226 411739 954625 456230 413923 642649 236445 413641 819485 372416 / 89 > 4¹⁰¹ [i]