Best Known (197−35, 197, s)-Nets in Base 3
(197−35, 197, 896)-Net over F3 — Constructive and digital
Digital (162, 197, 896)-net over F3, using
- 31 times duplication [i] based on digital (161, 196, 896)-net over F3, using
- t-expansion [i] based on digital (160, 196, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 49, 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, 49, 224)-net over F81, using
- t-expansion [i] based on digital (160, 196, 896)-net over F3, using
(197−35, 197, 4457)-Net over F3 — Digital
Digital (162, 197, 4457)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 4457, F3, 35) (dual of [4457, 4260, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 6605, F3, 35) (dual of [6605, 6408, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- 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(312, 44, F3, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3197, 6605, F3, 35) (dual of [6605, 6408, 36]-code), using
(197−35, 197, 1137198)-Net in Base 3 — Upper bound on s
There is no (162, 197, 1137199)-net in base 3, because
- 1 times m-reduction [i] would yield (162, 196, 1137199)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3279 204545 023073 331433 037482 246187 103330 216760 337995 598885 122640 325042 424184 460464 253499 475007 > 3196 [i]