MinT - Best Known (109, 109+77, s)-Nets in Base 4

Best Known (109, 109+77, s)-Nets in Base 4

(109, 109+77, 130)-Net over F₄ — Constructive and digital

Digital (109, 186, 130)-net over F₄, using

t-expansion [i] based on digital (105, 186, 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]

(109, 109+77, 245)-Net over F₄ — Digital

Digital (109, 186, 245)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+77, 4242)-Net in Base 4 — Upper bound on s

There is no (109, 186, 4243)-net in base 4, because

1 times m-reduction [i] would yield (109, 185, 4243)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2421 626486 974095 541603 453612 862245 680142 890873 916726 625446 086092 007193 591476 213517 608191 230093 130806 914951 801520 > 4¹⁸⁵ [i]