MinT - Best Known (174−103, 174, s)-Nets in Base 4

Best Known (174−103, 174, s)-Nets in Base 4

(174−103, 174, 66)-Net over F₄ — Constructive and digital

Digital (71, 174, 66)-net over F₄, using

t-expansion [i] based on digital (49, 174, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(174−103, 174, 105)-Net over F₄ — Digital

Digital (71, 174, 105)-net over F₄, using

t-expansion [i] based on digital (70, 174, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(174−103, 174, 688)-Net in Base 4 — Upper bound on s

There is no (71, 174, 689)-net in base 4, because

1 times m-reduction [i] would yield (71, 173, 689)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 149 667311 971186 095424 584337 549076 426695 886534 669236 324257 273637 116489 959168 537262 390960 024269 003295 465664 > 4¹⁷³ [i]