MinT - Best Known (243−171, 243, s)-Nets in Base 4

Best Known (243−171, 243, s)-Nets in Base 4

(243−171, 243, 66)-Net over F₄ — Constructive and digital

Digital (72, 243, 66)-net over F₄, using

t-expansion [i] based on digital (49, 243, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(243−171, 243, 105)-Net over F₄ — Digital

Digital (72, 243, 105)-net over F₄, using

t-expansion [i] based on digital (70, 243, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(243−171, 243, 492)-Net in Base 4 — Upper bound on s

There is no (72, 243, 493)-net in base 4, because

1 times m-reduction [i] would yield (72, 242, 493)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51 618731 501570 728626 027231 234600 646145 903544 309263 044450 055810 228527 886074 524917 963331 734311 723276 387235 099452 684150 003459 619656 510247 009692 386760 > 4²⁴² [i]