Best Known (106−22, 106, s)-Nets in Base 3
(106−22, 106, 600)-Net over F3 — Constructive and digital
Digital (84, 106, 600)-net over F3, using
- 32 times duplication [i] based on digital (82, 104, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 26, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 26, 150)-net over F81, using
(106−22, 106, 1310)-Net over F3 — Digital
Digital (84, 106, 1310)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3106, 1310, F3, 22) (dual of [1310, 1204, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 2214, F3, 22) (dual of [2214, 2108, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3106, 2214, F3, 22) (dual of [2214, 2108, 23]-code), using
(106−22, 106, 97196)-Net in Base 3 — Upper bound on s
There is no (84, 106, 97197)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 375 726411 802935 543415 588144 958126 393132 679399 527275 > 3106 [i]