MinT - Best Known (55, 55+93, s)-Nets in Base 3

Best Known (55, 55+93, s)-Nets in Base 3

(55, 55+93, 48)-Net over F₃ — Constructive and digital

Digital (55, 148, 48)-net over F₃, using

t-expansion [i] based on digital (45, 148, 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]

(55, 55+93, 64)-Net over F₃ — Digital

Digital (55, 148, 64)-net over F₃, using

t-expansion [i] based on digital (49, 148, 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]

(55, 55+93, 253)-Net over F₃ — Upper bound on s (digital)

There is no digital (55, 148, 254)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁸, 254, F₃, 93) (dual of [254, 106, 94]-code), but
- residual code [i] would yield OA(3⁵⁵, 160, S₃, 31), but
  - 1 times truncation [i] would yield OA(3⁵⁴, 159, S₃, 30), but
    - the linear programming bound shows that MÂ â‰¥ 57 512545 780273 008066 954941 653603 554048 685010 751297 100155 / 986015 082551 936588 824240 203061 > 3⁵⁴ [i]

(55, 55+93, 258)-Net in Base 3 — Upper bound on s

There is no (55, 148, 259)-net in base 3, because

1 times m-reduction [i] would yield (55, 147, 259)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15049 196965 340524 953206 127991 918151 092019 739165 314460 697766 641725 406989 > 3¹⁴⁷ [i]