MinT - Best Known (172−65, 172, s)-Nets in Base 4

Best Known (172−65, 172, s)-Nets in Base 4

(172−65, 172, 130)-Net over F₄ — Constructive and digital

Digital (107, 172, 130)-net over F₄, using

t-expansion [i] based on digital (105, 172, 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]

(172−65, 172, 311)-Net over F₄ — Digital

Digital (107, 172, 311)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−65, 172, 7004)-Net in Base 4 — Upper bound on s

There is no (107, 172, 7005)-net in base 4, because

1 times m-reduction [i] would yield (107, 171, 7005)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 964192 205777 215931 723371 203034 803600 147833 334071 015105 170066 560937 553136 311331 699058 045995 012973 693826 > 4¹⁷¹ [i]