Best Known (134, 134+29, s)-Nets in Base 3
(134, 134+29, 701)-Net over F3 — Constructive and digital
Digital (134, 163, 701)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (115, 144, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
- digital (5, 19, 13)-net over F3, using
(134, 134+29, 3956)-Net over F3 — Digital
Digital (134, 163, 3956)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3163, 3956, F3, 29) (dual of [3956, 3793, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3163, 6597, F3, 29) (dual of [6597, 6434, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3163, 6597, F3, 29) (dual of [6597, 6434, 30]-code), using
(134, 134+29, 1003220)-Net in Base 3 — Upper bound on s
There is no (134, 163, 1003221)-net in base 3, because
- 1 times m-reduction [i] would yield (134, 162, 1003221)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 196628 292161 046667 783180 127158 049477 494513 479216 247849 931991 849721 845343 852161 > 3162 [i]