Best Known (197−43, 197, s)-Nets in Base 3
(197−43, 197, 640)-Net over F3 — Constructive and digital
Digital (154, 197, 640)-net over F3, using
- 31 times duplication [i] based on digital (153, 196, 640)-net over F3, using
- t-expansion [i] based on digital (152, 196, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 49, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 49, 160)-net over F81, using
- t-expansion [i] based on digital (152, 196, 640)-net over F3, using
(197−43, 197, 1503)-Net over F3 — Digital
Digital (154, 197, 1503)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 1503, F3, 43) (dual of [1503, 1306, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using
- an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- discarding factors / shortening the dual code based on linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using
(197−43, 197, 123174)-Net in Base 3 — Upper bound on s
There is no (154, 197, 123175)-net in base 3, because
- 1 times m-reduction [i] would yield (154, 196, 123175)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3279 273401 141259 669407 862265 687493 928279 448967 552689 537352 477672 212937 109905 410929 222536 592231 > 3196 [i]