MinT - Best Known (78, 78+151, s)-Nets in Base 4

Best Known (78, 78+151, s)-Nets in Base 4

(78, 78+151, 104)-Net over F₄ — Constructive and digital

Digital (78, 229, 104)-net over F₄, using

t-expansion [i] based on digital (73, 229, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(78, 78+151, 112)-Net over F₄ — Digital

Digital (78, 229, 112)-net over F₄, using

t-expansion [i] based on digital (73, 229, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(78, 78+151, 588)-Net in Base 4 — Upper bound on s

There is no (78, 229, 589)-net in base 4, because

1 times m-reduction [i] would yield (78, 228, 589)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199595 256511 481426 998917 042766 540700 053078 462098 227408 450451 698468 397504 543369 188656 891542 878529 546312 697850 852955 977348 961752 814465 432892 > 4²²⁸ [i]