MinT - Best Known (62, 62+184, s)-Nets in Base 4

Best Known (62, 62+184, s)-Nets in Base 4

(62, 62+184, 66)-Net over F₄ — Constructive and digital

Digital (62, 246, 66)-net over F₄, using

t-expansion [i] based on digital (49, 246, 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+184, 99)-Net over F₄ — Digital

Digital (62, 246, 99)-net over F₄, using

t-expansion [i] based on digital (61, 246, 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+184, 259)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 246, 260)-net over F₄, because

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+184, 403)-Net in Base 4 — Upper bound on s

There is no (62, 246, 404)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13475 483952 346046 584840 481041 492032 547580 787745 824318 879400 470865 961055 901608 474213 839585 268705 066449 595553 449763 615182 613181 850568 274214 910543 461280 > 4²⁴⁶ [i]