MinT - Best Known (66, 66+188, s)-Nets in Base 4

Best Known (66, 66+188, s)-Nets in Base 4

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

Digital (66, 254, 66)-net over F₄, using

t-expansion [i] based on digital (49, 254, 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]

(66, 66+188, 99)-Net over F₄ — Digital

Digital (66, 254, 99)-net over F₄, using

t-expansion [i] based on digital (61, 254, 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]

(66, 66+188, 295)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 254, 296)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 296, F₄, 188) (dual of [296, 42, 189]-code), but
- residual code [i] would yield OA(4⁶⁶, 107, S₄, 47), but
  - the linear programming bound shows that MÂ â‰¥ 877667 617526 699604 115459 206661 960900 469804 406919 637481 696619 379668 118193 820628 156416 / 152 206215 688292 487856 759376 046017 491747 653675 > 4⁶⁶ [i]

(66, 66+188, 431)-Net in Base 4 — Upper bound on s

There is no (66, 254, 432)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 935 472184 524637 969723 044987 741383 919014 915428 449292 591230 525996 501212 048281 242130 726567 361450 032230 352323 270462 122628 430339 054913 238974 183308 666673 310940 > 4²⁵⁴ [i]