MinT - Best Known (38, 38+98, s)-Nets in Base 4

Best Known (38, 38+98, s)-Nets in Base 4

(38, 38+98, 56)-Net over F₄ — Constructive and digital

Digital (38, 136, 56)-net over F₄, using

t-expansion [i] based on digital (33, 136, 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]

(38, 38+98, 66)-Net over F₄ — Digital

Digital (38, 136, 66)-net over F₄, using

t-expansion [i] based on digital (37, 136, 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]

(38, 38+98, 243)-Net over F₄ — Upper bound on s (digital)

There is no digital (38, 136, 244)-net over F₄, because

2 times m-reduction [i] would yield digital (38, 134, 244)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁴, 244, F₄, 96) (dual of [244, 110, 97]-code), but
  - residual code [i] would yield OA(4³⁸, 147, S₄, 24), but
    - the linear programming bound shows that MÂ â‰¥ 5459 111453 470077 053881 598601 691985 346560 000000 / 69230 222284 404391 824599 > 4³⁸ [i]

(38, 38+98, 260)-Net in Base 4 — Upper bound on s

There is no (38, 136, 261)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8518 668417 414843 782393 092097 071660 233010 385773 078667 462244 139041 148522 480056 991744 > 4¹³⁶ [i]