MinT - Best Known (260−193, 260, s)-Nets in Base 4

Best Known (260−193, 260, s)-Nets in Base 4

(260−193, 260, 66)-Net over F₄ — Constructive and digital

Digital (67, 260, 66)-net over F₄, using

t-expansion [i] based on digital (49, 260, 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]

(260−193, 260, 99)-Net over F₄ — Digital

Digital (67, 260, 99)-net over F₄, using

t-expansion [i] based on digital (61, 260, 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]

(260−193, 260, 294)-Net over F₄ — Upper bound on s (digital)

There is no digital (67, 260, 295)-net over F₄, because

1 times m-reduction [i] would yield digital (67, 259, 295)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁹, 295, F₄, 192) (dual of [295, 36, 193]-code), but
  - residual code [i] would yield OA(4⁶⁷, 102, S₄, 48), but
    - the linear programming bound shows that MÂ â‰¥ 475693 928667 837098 050429 008442 895635 449460 006433 785673 218260 709740 118016 / 19 810806 413080 339108 130487 031223 > 4⁶⁷ [i]

(260−193, 260, 437)-Net in Base 4 — Upper bound on s

There is no (67, 260, 438)-net in base 4, because

1 times m-reduction [i] would yield (67, 259, 438)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 979357 351589 182330 443208 983981 479146 237194 918748 249577 527294 128449 602292 976708 967536 656930 735574 007683 064610 977422 971399 988205 366547 878171 398700 372655 304850 > 4²⁵⁹ [i]