MinT - Best Known (158−99, 158, s)-Nets in Base 3

Best Known (158−99, 158, s)-Nets in Base 3

(158−99, 158, 48)-Net over F₃ — Constructive and digital

Digital (59, 158, 48)-net over F₃, using

t-expansion [i] based on digital (45, 158, 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]

(158−99, 158, 64)-Net over F₃ — Digital

Digital (59, 158, 64)-net over F₃, using

t-expansion [i] based on digital (49, 158, 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]

(158−99, 158, 273)-Net over F₃ — Upper bound on s (digital)

There is no digital (59, 158, 274)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁵⁸, 274, F₃, 99) (dual of [274, 116, 100]-code), but
- residual code [i] would yield OA(3⁵⁹, 174, S₃, 33), but
  - 1 times truncation [i] would yield OA(3⁵⁸, 173, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 47 314229 835710 776345 670615 418214 790563 428291 670976 086718 521800 / 9965 095822 320687 546955 148607 190123 > 3⁵⁸ [i]

(158−99, 158, 277)-Net in Base 3 — Upper bound on s

There is no (59, 158, 278)-net in base 3, because

1 times m-reduction [i] would yield (59, 157, 278)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 918 630549 308044 297780 744178 078317 140082 437771 670421 945847 416558 991712 751565 > 3¹⁵⁷ [i]