MinT - Best Known (232−170, 232, s)-Nets in Base 4

Best Known (232−170, 232, s)-Nets in Base 4

Digital (62, 232, 66)-net over F₄, using

t-expansion [i] based on digital (49, 232, 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]

Digital (62, 232, 99)-net over F₄, using

t-expansion [i] based on digital (61, 232, 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]

There is no digital (62, 232, 319)-net over F₄, because

2 times m-reduction [i] would yield digital (62, 230, 319)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁰, 319, F₄, 168) (dual of [319, 89, 169]-code), but
  - residual code [i] would yield OA(4⁶², 150, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 1 659762 510567 225532 778073 956682 851160 927584 506340 544111 988906 878425 443399 317508 383360 785706 508045 270473 316579 147776 / 70196 277944 430470 237066 876146 512897 904291 630830 313677 137903 195390 483423 677609 > 4⁶² [i]

There is no (62, 232, 410)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56 610683 071390 878694 105468 100557 978983 351322 367850 504095 643351 031364 264578 472545 244559 438286 935774 795357 472030 569551 867652 955268 553658 421020 > 4²³² [i]