MinT - Best Known (138−100, 138, s)-Nets in Base 4

Best Known (138−100, 138, s)-Nets in Base 4

(138−100, 138, 56)-Net over F₄ — Constructive and digital

Digital (38, 138, 56)-net over F₄, using

t-expansion [i] based on digital (33, 138, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(138−100, 138, 66)-Net over F₄ — Digital

Digital (38, 138, 66)-net over F₄, using

t-expansion [i] based on digital (37, 138, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

(138−100, 138, 225)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 138, 226)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹³⁸, 226, F₄, 100) (dual of [226, 88, 101]-code), but
- residual code [i] would yield OA(4³⁸, 125, S₄, 25), but
  - the linear programming bound shows that MÂ â‰¥ 389 567034 474678 876680 715278 147118 516148 790886 400000 / 5089 117647 497197 040645 830161 > 4³⁸ [i]

(138−100, 138, 258)-Net in Base 4 — Upper bound on s

There is no (38, 138, 259)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 124078 824661 847265 693320 957681 772325 034369 342684 079627 064728 551494 034886 366303 729616 > 4¹³⁸ [i]