MinT - Best Known (68, 68+168, s)-Nets in Base 4

Best Known (68, 68+168, s)-Nets in Base 4

(68, 68+168, 66)-Net over F₄ — Constructive and digital

Digital (68, 236, 66)-net over F₄, using

t-expansion [i] based on digital (49, 236, 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]

(68, 68+168, 99)-Net over F₄ — Digital

Digital (68, 236, 99)-net over F₄, using

t-expansion [i] based on digital (61, 236, 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]

(68, 68+168, 414)-Net over F₄ — Upper bound on s (digital)

There is no digital (68, 236, 415)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁶, 415, F₄, 168) (dual of [415, 179, 169]-code), but
- residual code [i] would yield OA(4⁶⁸, 246, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 8 266893 710032 655425 717892 686988 538675 685493 888138 140590 751265 026314 680838 406117 785600 000000 / 94 513486 949920 285641 128495 398550 971508 431933 193771 > 4⁶⁸ [i]

(68, 68+168, 459)-Net in Base 4 — Upper bound on s

There is no (68, 236, 460)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13680 596066 044375 535151 374162 205652 826688 794651 472320 913666 734103 704128 007102 861674 912149 574887 089132 214328 712966 613319 585637 045704 651279 746544 > 4²³⁶ [i]