Best Known (206−138, 206, s)-Nets in Base 3
(206−138, 206, 48)-Net over F3 — Constructive and digital
Digital (68, 206, 48)-net over F3, using
- t-expansion [i] based on digital (45, 206, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(206−138, 206, 72)-Net over F3 — Digital
Digital (68, 206, 72)-net over F3, using
- t-expansion [i] based on digital (67, 206, 72)-net over F3, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 67 and N(F) ≥ 72, using
- net from sequence [i] based on digital (67, 71)-sequence over F3, using
(206−138, 206, 213)-Net over F3 — Upper bound on s (digital)
There is no digital (68, 206, 214)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3206, 214, F3, 138) (dual of [214, 8, 139]-code), but
- residual code [i] would yield linear OA(368, 75, F3, 46) (dual of [75, 7, 47]-code), but
- 1 times truncation [i] would yield linear OA(367, 74, F3, 45) (dual of [74, 7, 46]-code), but
- residual code [i] would yield linear OA(322, 28, F3, 15) (dual of [28, 6, 16]-code), but
- “HHM†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(322, 28, F3, 15) (dual of [28, 6, 16]-code), but
- 1 times truncation [i] would yield linear OA(367, 74, F3, 45) (dual of [74, 7, 46]-code), but
- residual code [i] would yield linear OA(368, 75, F3, 46) (dual of [75, 7, 47]-code), but
(206−138, 206, 288)-Net in Base 3 — Upper bound on s
There is no (68, 206, 289)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 194 923645 359953 235602 694963 439704 796209 344867 341817 851808 015356 475384 115984 035474 314289 340445 158963 > 3206 [i]