MinT - Best Known (67, 67+168, s)-Nets in Base 4

Best Known (67, 67+168, s)-Nets in Base 4

(67, 67+168, 66)-Net over F₄ — Constructive and digital

Digital (67, 235, 66)-net over F₄, using

t-expansion [i] based on digital (49, 235, 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]

(67, 67+168, 99)-Net over F₄ — Digital

Digital (67, 235, 99)-net over F₄, using

t-expansion [i] based on digital (61, 235, 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]

(67, 67+168, 398)-Net over F₄ — Upper bound on s (digital)

There is no digital (67, 235, 399)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁵, 399, F₄, 168) (dual of [399, 164, 169]-code), but
- residual code [i] would yield OA(4⁶⁷, 230, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 133 864877 868223 848277 340960 330245 263082 378026 507294 938785 509330 612908 753764 170596 352000 / 6063 223087 091274 607869 862566 805531 022255 644183 > 4⁶⁷ [i]

(67, 67+168, 450)-Net in Base 4 — Upper bound on s

There is no (67, 235, 451)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3184 087577 669098 332861 003083 671804 045888 510850 284977 675772 290630 065289 140720 161063 928595 753020 437893 871936 297866 857831 038873 597018 142680 267168 > 4²³⁵ [i]