MinT - Best Known (62, 62+198, s)-Nets in Base 4

Best Known (62, 62+198, s)-Nets in Base 4

(62, 62+198, 66)-Net over F₄ — Constructive and digital

Digital (62, 260, 66)-net over F₄, using

t-expansion [i] based on digital (49, 260, 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+198, 99)-Net over F₄ — Digital

Digital (62, 260, 99)-net over F₄, using

t-expansion [i] based on digital (61, 260, 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+198, 257)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 260, 258)-net over F₄, because

6 times m-reduction [i] would yield digital (62, 254, 258)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 258, F₄, 192) (dual of [258, 4, 193]-code), but
  - Griesmer bound [i]

(62, 62+198, 263)-Net in Base 4 — Upper bound on s

There is no (62, 260, 264)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4²⁶⁰, 264, S₄, 198), but
- the (dual) Plotkin bound shows that MÂ â‰¥ 878 694100 496718 043517 683302 282418 331810 487718 418343 092402 491322 775749 527474 899974 671687 634004 666183 037093 927858 109549 828751 614463 963730 408009 475621 262727 315456 / 199 > 4²⁶⁰ [i]