MinT - Best Known (243−123, 243, s)-Nets in Base 4

Best Known (243−123, 243, s)-Nets in Base 4

(243−123, 243, 130)-Net over F₄ — Constructive and digital

Digital (120, 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]

(243−123, 243, 168)-Net over F₄ — Digital

Digital (120, 243, 168)-net over F₄, using

t-expansion [i] based on digital (115, 243, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(243−123, 243, 1871)-Net in Base 4 — Upper bound on s

There is no (120, 243, 1872)-net in base 4, because

1 times m-reduction [i] would yield (120, 242, 1872)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 768548 205191 864551 695325 025004 290135 101164 019884 061671 609342 756141 766285 320502 795028 683740 434986 024767 238193 448000 810167 614684 890630 936485 148332 > 4²⁴² [i]