MinT - Best Known (243−135, 243, s)-Nets in Base 4

Best Known (243−135, 243, s)-Nets in Base 4

(243−135, 243, 130)-Net over F₄ — Constructive and digital

Digital (108, 243, 130)-net over F₄, using

t-expansion [i] based on digital (105, 243, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(243−135, 243, 144)-Net over F₄ — Digital

Digital (108, 243, 144)-net over F₄, using

t-expansion [i] based on digital (91, 243, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(243−135, 243, 1230)-Net in Base 4 — Upper bound on s

There is no (108, 243, 1231)-net in base 4, because

1 times m-reduction [i] would yield (108, 242, 1231)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51 491720 336101 878638 117335 376688 632221 416584 633405 816303 155340 217441 727523 286177 525362 427746 996002 631042 938844 585743 921093 212237 648114 616818 416728 > 4²⁴² [i]