MinT - Best Known (253−175, 253, s)-Nets in Base 4

Best Known (253−175, 253, s)-Nets in Base 4

(253−175, 253, 104)-Net over F₄ — Constructive and digital

Digital (78, 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]

(253−175, 253, 112)-Net over F₄ — Digital

Digital (78, 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]

(253−175, 253, 544)-Net in Base 4 — Upper bound on s

There is no (78, 253, 545)-net in base 4, because

1 times m-reduction [i] would yield (78, 252, 545)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56 788708 842099 752319 171452 711503 886537 608688 397299 741775 398253 970214 201284 803978 484728 586347 716149 801124 662211 691479 741398 896953 310413 161269 468510 503680 > 4²⁵² [i]