MinT - Best Known (178−158, 178, s)-Nets in Base 4

Best Known (178−158, 178, s)-Nets in Base 4

(178−158, 178, 33)-Net over F₄ — Constructive and digital

Digital (20, 178, 33)-net over F₄, using

t-expansion [i] based on digital (15, 178, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(178−158, 178, 41)-Net over F₄ — Digital

Digital (20, 178, 41)-net over F₄, using

t-expansion [i] based on digital (18, 178, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(178−158, 178, 75)-Net in Base 4 — Upper bound on s

There is no (20, 178, 76)-net in base 4, because

31 times m-reduction [i] would yield (20, 147, 76)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁴⁷, 76, S₄, 2, 127), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 33320 656839 455705 024410 610625 836996 286730 562103 422297 068511 086451 841686 299839 736160 714752 > 4¹⁴⁷ [i]