MinT - Best Known (228−164, 228, s)-Nets in Base 4

Best Known (228−164, 228, s)-Nets in Base 4

(228−164, 228, 66)-Net over F₄ — Constructive and digital

Digital (64, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 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]

(228−164, 228, 99)-Net over F₄ — Digital

Digital (64, 228, 99)-net over F₄, using

t-expansion [i] based on digital (61, 228, 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]

(228−164, 228, 367)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 228, 368)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁸, 368, F₄, 164) (dual of [368, 140, 165]-code), but
- residual code [i] would yield OA(4⁶⁴, 203, S₄, 41), but
  - the linear programming bound shows that MÂ â‰¥ 9237 828678 014661 888733 201438 074087 629177 850315 553208 766694 988543 356653 154675 278877 490717 655040 / 25 848321 162508 053807 967718 560003 483887 415061 992757 599441 > 4⁶⁴ [i]

(228−164, 228, 428)-Net in Base 4 — Upper bound on s

There is no (64, 228, 429)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 198340 249794 106406 917576 000483 680492 919732 604998 436269 224575 353617 755262 640599 946279 539191 642689 646598 401580 758802 001078 836344 232709 177000 > 4²²⁸ [i]