Best Known (179, 179+40, s)-Nets in Base 3
(179, 179+40, 896)-Net over F3 — Constructive and digital
Digital (179, 219, 896)-net over F3, using
- t-expansion [i] based on digital (178, 219, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- 1 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
(179, 179+40, 4065)-Net over F3 — Digital
Digital (179, 219, 4065)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3219, 4065, F3, 40) (dual of [4065, 3846, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 6597, F3, 40) (dual of [6597, 6378, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3219, 6597, F3, 40) (dual of [6597, 6378, 41]-code), using
(179, 179+40, 696212)-Net in Base 3 — Upper bound on s
There is no (179, 219, 696213)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 308 717421 091985 623021 818045 467928 764584 845405 233212 242209 290261 587693 061467 921571 560417 541297 162567 123089 > 3219 [i]