MinT - Best Known (178−69, 178, s)-Nets in Base 4

Best Known (178−69, 178, s)-Nets in Base 4

(178−69, 178, 130)-Net over F₄ — Constructive and digital

Digital (109, 178, 130)-net over F₄, using

t-expansion [i] based on digital (105, 178, 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]

(178−69, 178, 293)-Net over F₄ — Digital

Digital (109, 178, 293)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−69, 178, 6118)-Net in Base 4 — Upper bound on s

There is no (109, 178, 6119)-net in base 4, because

1 times m-reduction [i] would yield (109, 177, 6119)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36812 131937 238891 906061 763512 065519 848760 205251 510102 457969 363202 369599 551373 167148 113512 903869 626777 938317 > 4¹⁷⁷ [i]