Best Known (228−41, 228, s)-Nets in Base 3
(228−41, 228, 1480)-Net over F3 — Constructive and digital
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
(228−41, 228, 4571)-Net over F3 — Digital
Digital (187, 228, 4571)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3228, 4571, F3, 41) (dual of [4571, 4343, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 6602, F3, 41) (dual of [6602, 6374, 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(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- 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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 6602, F3, 41) (dual of [6602, 6374, 42]-code), using
(228−41, 228, 1080425)-Net in Base 3 — Upper bound on s
There is no (187, 228, 1080426)-net in base 3, because
- 1 times m-reduction [i] would yield (187, 227, 1080426)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 025501 623671 850625 526803 111684 973959 940246 348614 247332 021170 142323 456647 040649 880310 485547 754042 845106 079561 > 3227 [i]