MinT - Best Known (138−72, 138, s)-Nets in Base 4

Best Known (138−72, 138, s)-Nets in Base 4

(138−72, 138, 66)-Net over F₄ — Constructive and digital

Digital (66, 138, 66)-net over F₄, using

t-expansion [i] based on digital (49, 138, 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]

(138−72, 138, 99)-Net over F₄ — Digital

Digital (66, 138, 99)-net over F₄, using

t-expansion [i] based on digital (61, 138, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(138−72, 138, 938)-Net in Base 4 — Upper bound on s

There is no (66, 138, 939)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 125873 898266 084876 320264 449303 956076 827053 171245 553999 212269 675898 856041 329696 958850 > 4¹³⁸ [i]