MinT - Best Known (258−196, 258, s)-Nets in Base 4

Best Known (258−196, 258, s)-Nets in Base 4

(258−196, 258, 66)-Net over F₄ — Constructive and digital

Digital (62, 258, 66)-net over F₄, using

t-expansion [i] based on digital (49, 258, 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]

(258−196, 258, 99)-Net over F₄ — Digital

Digital (62, 258, 99)-net over F₄, using

t-expansion [i] based on digital (61, 258, 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]

(258−196, 258, 257)-Net over F₄ — Upper bound on s (digital)

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

4 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]

(258−196, 258, 401)-Net in Base 4 — Upper bound on s

There is no (62, 258, 402)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 249769 619373 631550 786066 196862 228055 931082 596561 781705 567714 545536 849476 928170 973688 359004 747583 957494 153552 040099 135804 627649 191503 025271 770440 518811 933184 > 4²⁵⁸ [i]