MinT - Best Known (251−188, 251, s)-Nets in Base 4

Best Known (251−188, 251, s)-Nets in Base 4

(251−188, 251, 66)-Net over F₄ — Constructive and digital

Digital (63, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 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]

(251−188, 251, 99)-Net over F₄ — Digital

Digital (63, 251, 99)-net over F₄, using

t-expansion [i] based on digital (61, 251, 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]

(251−188, 251, 263)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 251, 264)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵¹, 264, F₄, 188) (dual of [264, 13, 189]-code), but
- residual code [i] would yield OA(4⁶³, 75, S₄, 47), but
  - the linear programming bound shows that MÂ â‰¥ 108 051901 662607 115934 306119 345453 950792 892416 / 1 213839 > 4⁶³ [i]

(251−188, 251, 409)-Net in Base 4 — Upper bound on s

There is no (63, 251, 410)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 793919 005454 916947 398759 857137 380559 326335 892222 370395 866005 895168 434883 037002 335406 610882 123746 248911 254521 765419 450541 630128 968102 229650 477878 001984 > 4²⁵¹ [i]