MinT - Best Known (78, 78+171, s)-Nets in Base 4

Best Known (78, 78+171, s)-Nets in Base 4

(78, 78+171, 104)-Net over F₄ — Constructive and digital

Digital (78, 249, 104)-net over F₄, using

t-expansion [i] based on digital (73, 249, 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]

(78, 78+171, 112)-Net over F₄ — Digital

Digital (78, 249, 112)-net over F₄, using

t-expansion [i] based on digital (73, 249, 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]

(78, 78+171, 550)-Net in Base 4 — Upper bound on s

There is no (78, 249, 551)-net in base 4, because

1 times m-reduction [i] would yield (78, 248, 551)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 232600 957273 603858 666546 909919 614307 091183 725521 991526 886748 511926 444609 233614 636015 917936 817738 520950 234260 794362 895106 473408 457312 182585 573188 018400 > 4²⁴⁸ [i]