MinT - Best Known (258−192, 258, s)-Nets in Base 4

Best Known (258−192, 258, s)-Nets in Base 4

(258−192, 258, 66)-Net over F₄ — Constructive and digital

Digital (66, 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]

(258−192, 258, 99)-Net over F₄ — Digital

Digital (66, 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]

(258−192, 258, 282)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 258, 283)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁸, 283, F₄, 192) (dual of [283, 25, 193]-code), but
- residual code [i] would yield OA(4⁶⁶, 90, S₄, 48), but
  - the linear programming bound shows that MÂ â‰¥ 15992 796442 957684 354900 601418 173389 061260 037338 242442 330112 / 2 261259 518366 924075 > 4⁶⁶ [i]

(258−192, 258, 430)-Net in Base 4 — Upper bound on s

There is no (66, 258, 431)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 259214 051897 912997 530198 240714 034287 254184 368097 574624 788349 571831 929571 607589 512444 162100 714728 674582 966541 957229 438322 174255 477687 729717 714527 766843 708510 > 4²⁵⁸ [i]