MinT - Best Known (257−182, 257, s)-Nets in Base 4

Best Known (257−182, 257, s)-Nets in Base 4

(257−182, 257, 104)-Net over F₄ — Constructive and digital

Digital (75, 257, 104)-net over F₄, using

t-expansion [i] based on digital (73, 257, 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]

(257−182, 257, 112)-Net over F₄ — Digital

Digital (75, 257, 112)-net over F₄, using

t-expansion [i] based on digital (73, 257, 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]

(257−182, 257, 507)-Net in Base 4 — Upper bound on s

There is no (75, 257, 508)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 55968 337940 078850 943577 799685 917350 199573 599323 788491 319171 968672 222116 783091 003646 582147 410415 031978 954918 233026 338907 969624 772805 694405 199404 856615 329425 > 4²⁵⁷ [i]