MinT - Best Known (62, 62+186, s)-Nets in Base 4

Best Known (62, 62+186, s)-Nets in Base 4

(62, 62+186, 66)-Net over F₄ — Constructive and digital

Digital (62, 248, 66)-net over F₄, using

t-expansion [i] based on digital (49, 248, 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]

(62, 62+186, 99)-Net over F₄ — Digital

Digital (62, 248, 99)-net over F₄, using

t-expansion [i] based on digital (61, 248, 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]

(62, 62+186, 259)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 248, 260)-net over F₄, because

2 times m-reduction [i] would yield digital (62, 246, 260)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁶, 260, F₄, 184) (dual of [260, 14, 185]-code), but
  - residual code [i] would yield linear OA(4⁶², 75, F₄, 46) (dual of [75, 13, 47]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(62, 62+186, 403)-Net in Base 4 — Upper bound on s

There is no (62, 248, 404)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 239436 795379 470654 721272 434594 663753 414926 757861 317220 859147 549870 476674 783779 078350 639316 444200 524304 739490 164287 928512 519002 240959 635274 627125 583904 > 4²⁴⁸ [i]