Best Known (188, 188+41, s)-Nets in Base 3
(188, 188+41, 1480)-Net over F3 — Constructive and digital
Digital (188, 229, 1480)-net over F3, using
- 31 times duplication [i] based on digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
(188, 188+41, 4702)-Net over F3 — Digital
Digital (188, 229, 4702)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3229, 4702, F3, 41) (dual of [4702, 4473, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 6605, F3, 41) (dual of [6605, 6376, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- 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(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(40) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 6605, F3, 41) (dual of [6605, 6376, 42]-code), using
(188, 188+41, 1141434)-Net in Base 3 — Upper bound on s
There is no (188, 229, 1141435)-net in base 3, because
- 1 times m-reduction [i] would yield (188, 228, 1141435)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 076414 436174 895479 610012 585893 488567 203769 488650 614810 568873 553550 242530 142688 516237 024402 250777 230082 809017 > 3228 [i]