MinT - Best Known (179−117, 179, s)-Nets in Base 3

Best Known (179−117, 179, s)-Nets in Base 3

(179−117, 179, 48)-Net over F₃ — Constructive and digital

Digital (62, 179, 48)-net over F₃, using

t-expansion [i] based on digital (45, 179, 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]

(179−117, 179, 64)-Net over F₃ — Digital

Digital (62, 179, 64)-net over F₃, using

t-expansion [i] based on digital (49, 179, 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]

(179−117, 179, 216)-Net over F₃ — Upper bound on s (digital)

There is no digital (62, 179, 217)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁷⁹, 217, F₃, 117) (dual of [217, 38, 118]-code), but
- residual code [i] would yield OA(3⁶², 99, S₃, 39), but
  - the linear programming bound shows that MÂ â‰¥ 8341 316354 832968 167115 756462 681271 298651 307150 960469 759639 / 19004 191492 894766 602246 400000 > 3⁶² [i]

(179−117, 179, 273)-Net in Base 3 — Upper bound on s

There is no (62, 179, 274)-net in base 3, because

1 times m-reduction [i] would yield (62, 178, 274)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 614246 464583 790311 922100 896400 186683 075990 220290 527799 694874 221774 947109 727506 857765 > 3¹⁷⁸ [i]