MinT - Best Known (81, 81+178, s)-Nets in Base 4

Best Known (81, 81+178, s)-Nets in Base 4

(81, 81+178, 104)-Net over F₄ — Constructive and digital

Digital (81, 259, 104)-net over F₄, using

t-expansion [i] based on digital (73, 259, 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]

(81, 81+178, 129)-Net over F₄ — Digital

Digital (81, 259, 129)-net over F₄, using

net from sequence [i] based on digital (81, 128)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
  - Drinfeld modules of rank 1 [i]

(81, 81+178, 568)-Net in Base 4 — Upper bound on s

There is no (81, 259, 569)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 939399 586046 098352 953141 940106 552524 427315 428699 851412 949406 952508 231996 320679 757207 221556 145099 565052 840899 934394 496729 311609 295282 196365 306272 780494 077280 > 4²⁵⁹ [i]