MinT - Best Known (39, 138, s)-Nets in Base 4

Best Known (39, 138, s)-Nets in Base 4

(39, 138, 56)-Net over F₄ — Constructive and digital

Digital (39, 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]

(39, 138, 66)-Net over F₄ — Digital

Digital (39, 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]

(39, 138, 260)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 138, 261)-net over F₄, because

3 times m-reduction [i] would yield digital (39, 135, 261)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁵, 261, F₄, 96) (dual of [261, 126, 97]-code), but
  - residual code [i] would yield OA(4³⁹, 164, S₄, 24), but
    - the linear programming bound shows that MÂ â‰¥ 253 017119 292002 552737 316772 682392 509261 807616 / 821 395315 909793 005123 > 4³⁹ [i]

(39, 138, 268)-Net in Base 4 — Upper bound on s

There is no (39, 138, 269)-net in base 4, because

1 times m-reduction [i] would yield (39, 137, 269)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31250 348258 016491 801873 746197 724599 257479 811515 037536 983092 933443 675480 873811 734144 > 4¹³⁷ [i]