Best Known (102−25, 102, s)-Nets in Base 3
(102−25, 102, 328)-Net over F3 — Constructive and digital
Digital (77, 102, 328)-net over F3, using
- 32 times duplication [i] based on digital (75, 100, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 25, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 25, 82)-net over F81, using
(102−25, 102, 567)-Net over F3 — Digital
Digital (77, 102, 567)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3102, 567, F3, 25) (dual of [567, 465, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3102, 728, F3, 25) (dual of [728, 626, 26]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(3102, 728, F3, 25) (dual of [728, 626, 26]-code), using
(102−25, 102, 27410)-Net in Base 3 — Upper bound on s
There is no (77, 102, 27411)-net in base 3, because
- 1 times m-reduction [i] would yield (77, 101, 27411)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 546347 659990 086843 381083 304768 124232 439237 879113 > 3101 [i]