MinT - Best Known (220−160, 220, s)-Nets in Base 4

Best Known (220−160, 220, s)-Nets in Base 4

(220−160, 220, 66)-Net over F₄ — Constructive and digital

Digital (60, 220, 66)-net over F₄, using

t-expansion [i] based on digital (49, 220, 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]

(220−160, 220, 91)-Net over F₄ — Digital

Digital (60, 220, 91)-net over F₄, using

t-expansion [i] based on digital (50, 220, 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]

(220−160, 220, 319)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 220, 320)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁰, 320, F₄, 160) (dual of [320, 100, 161]-code), but
- residual code [i] would yield OA(4⁶⁰, 159, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 89256 467878 907972 172600 180176 154060 802582 684769 875981 969312 131469 276806 223240 396340 011634 339610 624000 / 64975 225388 099566 116025 141010 137689 743517 140250 845468 320822 687201 > 4⁶⁰ [i]

(220−160, 220, 398)-Net in Base 4 — Upper bound on s

There is no (60, 220, 399)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 959592 616138 307097 234881 585150 878241 708776 575004 219938 836080 109130 475607 922769 243013 255784 200582 883301 296938 038641 091320 020273 831624 > 4²²⁰ [i]