MinT - Best Known (249−127, 249, s)-Nets in Base 4

Best Known (249−127, 249, s)-Nets in Base 4

(249−127, 249, 130)-Net over F₄ — Constructive and digital

Digital (122, 249, 130)-net over F₄, using

t-expansion [i] based on digital (105, 249, 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]

(249−127, 249, 168)-Net over F₄ — Digital

Digital (122, 249, 168)-net over F₄, using

t-expansion [i] based on digital (115, 249, 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]

(249−127, 249, 1847)-Net in Base 4 — Upper bound on s

There is no (122, 249, 1848)-net in base 4, because

1 times m-reduction [i] would yield (122, 248, 1848)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206613 328655 348404 308131 541660 431907 217316 469773 719192 702660 591436 313120 133216 947666 657648 783462 006049 127263 553453 740261 603677 531881 955408 618026 917600 > 4²⁴⁸ [i]