MinT - Best Known (58, 58+144, s)-Nets in Base 4

Best Known (58, 58+144, s)-Nets in Base 4

(58, 58+144, 66)-Net over F₄ — Constructive and digital

Digital (58, 202, 66)-net over F₄, using

t-expansion [i] based on digital (49, 202, 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]

(58, 58+144, 91)-Net over F₄ — Digital

Digital (58, 202, 91)-net over F₄, using

t-expansion [i] based on digital (50, 202, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(58, 58+144, 357)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 202, 358)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰², 358, F₄, 144) (dual of [358, 156, 145]-code), but
- residual code [i] would yield OA(4⁵⁸, 213, S₄, 36), but
  - the linear programming bound shows that MÂ â‰¥ 7 866958 687285 876635 384332 274577 437607 350342 946368 588838 836611 539357 984012 369920 / 88 856238 935032 449112 554626 779870 493844 853867 > 4⁵⁸ [i]

(58, 58+144, 393)-Net in Base 4 — Upper bound on s

There is no (58, 202, 394)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44 176468 826956 372226 063154 688336 026855 587329 197193 658178 694184 608151 113806 116344 344373 837135 457779 323885 533487 713189 000770 > 4²⁰² [i]