Best Known (122−21, 122, s)-Nets in Base 3
(122−21, 122, 692)-Net over F3 — Constructive and digital
Digital (101, 122, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (91, 112, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
- digital (0, 10, 4)-net over F3, using
(122−21, 122, 4314)-Net over F3 — Digital
Digital (101, 122, 4314)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3122, 4314, F3, 21) (dual of [4314, 4192, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 6602, F3, 21) (dual of [6602, 6480, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3121, 6601, F3, 21) (dual of [6601, 6480, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3121, 6601, F3, 21) (dual of [6601, 6480, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 6602, F3, 21) (dual of [6602, 6480, 22]-code), using
(122−21, 122, 1343106)-Net in Base 3 — Upper bound on s
There is no (101, 122, 1343107)-net in base 3, because
- 1 times m-reduction [i] would yield (101, 121, 1343107)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5391 055061 387790 819709 314252 860594 991677 429655 077082 899541 > 3121 [i]