MinT - Best Known (223−160, 223, s)-Nets in Base 4

Best Known (223−160, 223, s)-Nets in Base 4

(223−160, 223, 66)-Net over F₄ — Constructive and digital

Digital (63, 223, 66)-net over F₄, using

t-expansion [i] based on digital (49, 223, 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]

(223−160, 223, 99)-Net over F₄ — Digital

Digital (63, 223, 99)-net over F₄, using

t-expansion [i] based on digital (61, 223, 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]

(223−160, 223, 368)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 223, 369)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²³, 369, F₄, 160) (dual of [369, 146, 161]-code), but
- residual code [i] would yield OA(4⁶³, 208, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 5 853210 937914 115965 830373 633983 054380 602394 595992 687313 196476 039443 785703 042842 624000 / 66248 338389 800632 875754 408934 650630 828021 253383 > 4⁶³ [i]

(223−160, 223, 423)-Net in Base 4 — Upper bound on s

There is no (63, 223, 424)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206 349161 011347 351464 571179 956719 500385 019927 530646 350640 847995 723864 518363 157374 235447 157332 377385 186641 613910 797978 948127 302125 600268 > 4²²³ [i]