MinT - Best Known (75, 75+178, s)-Nets in Base 4

Best Known (75, 75+178, s)-Nets in Base 4

(75, 75+178, 104)-Net over F₄ — Constructive and digital

Digital (75, 253, 104)-net over F₄, using

t-expansion [i] based on digital (73, 253, 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]

(75, 75+178, 112)-Net over F₄ — Digital

Digital (75, 253, 112)-net over F₄, using

t-expansion [i] based on digital (73, 253, 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]

(75, 75+178, 511)-Net in Base 4 — Upper bound on s

There is no (75, 253, 512)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 221 986160 106438 560104 086793 841398 054014 672170 241562 312645 858383 911656 786063 248518 996755 605338 307250 890089 922636 538585 173210 452637 056972 222517 350454 902473 > 4²⁵³ [i]