Best Known (73, 73+22, s)-Nets in Base 3
(73, 73+22, 400)-Net over F3 — Constructive and digital
Digital (73, 95, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (73, 96, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 24, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 24, 100)-net over F81, using
(73, 73+22, 708)-Net over F3 — Digital
Digital (73, 95, 708)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(395, 708, F3, 22) (dual of [708, 613, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(395, 763, F3, 22) (dual of [763, 668, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(310, 34, F3, 5) (dual of [34, 24, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(395, 763, F3, 22) (dual of [763, 668, 23]-code), using
(73, 73+22, 32391)-Net in Base 3 — Upper bound on s
There is no (73, 95, 32392)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2120 985678 555609 269296 672095 709809 221369 069985 > 395 [i]