MinT - Best Known (258−184, 258, s)-Nets in Base 4

Best Known (258−184, 258, s)-Nets in Base 4

(258−184, 258, 104)-Net over F₄ — Constructive and digital

Digital (74, 258, 104)-net over F₄, using

t-expansion [i] based on digital (73, 258, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(258−184, 258, 112)-Net over F₄ — Digital

Digital (74, 258, 112)-net over F₄, using

t-expansion [i] based on digital (73, 258, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(258−184, 258, 447)-Net over F₄ — Upper bound on s (digital)

There is no digital (74, 258, 448)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁸, 448, F₄, 184) (dual of [448, 190, 185]-code), but
- residual code [i] would yield OA(4⁷⁴, 263, S₄, 46), but
  - the linear programming bound shows that MÂ â‰¥ 13 130009 898078 888671 716095 039721 251826 238799 418000 706327 723838 321723 415169 139087 202337 533267 542016 / 26961 506472 027069 344898 146639 493268 320567 982955 773129 > 4⁷⁴ [i]

(258−184, 258, 497)-Net in Base 4 — Upper bound on s

There is no (74, 258, 498)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 239221 903952 725071 730277 854721 629077 126041 406211 980356 347658 866213 728885 536453 488031 276974 477701 686621 465123 557152 926723 670563 860553 295623 886583 189044 644488 > 4²⁵⁸ [i]