MinT - Best Known (142−93, 142, s)-Nets in Base 3

Best Known (142−93, 142, s)-Nets in Base 3

(142−93, 142, 48)-Net over F₃ — Constructive and digital

Digital (49, 142, 48)-net over F₃, using

t-expansion [i] based on digital (45, 142, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(142−93, 142, 64)-Net over F₃ — Digital

Digital (49, 142, 64)-net over F₃, using

net from sequence [i] based on digital (49, 63)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
  - a curve from the manYPoints database [i]

(142−93, 142, 178)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 142, 179)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴², 179, F₃, 93) (dual of [179, 37, 94]-code), but
- residual code [i] would yield OA(3⁴⁹, 85, S₃, 31), but
  - the linear programming bound shows that MÂ â‰¥ 680029 495837 431892 373726 566388 206026 660443 378717 028919 971748 021750 421500 256283 / 2 775791 539437 123450 845959 961813 360302 598435 660143 224610 > 3⁴⁹ [i]

(142−93, 142, 218)-Net in Base 3 — Upper bound on s

There is no (49, 142, 219)-net in base 3, because

1 times m-reduction [i] would yield (49, 141, 219)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 20 368334 404390 959451 562640 977170 379786 696898 223629 539621 222167 819773 > 3¹⁴¹ [i]