Best Known (170, 215, s)-Nets in Base 3
(170, 215, 688)-Net over F3 — Constructive and digital
Digital (170, 215, 688)-net over F3, using
- t-expansion [i] based on digital (169, 215, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 54, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
(170, 215, 1960)-Net over F3 — Digital
Digital (170, 215, 1960)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3215, 1960, F3, 45) (dual of [1960, 1745, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 2207, F3, 45) (dual of [2207, 1992, 46]-code), using
- construction XX applied to Ce(45) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(45) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3215, 2207, F3, 45) (dual of [2207, 1992, 46]-code), using
(170, 215, 198090)-Net in Base 3 — Upper bound on s
There is no (170, 215, 198091)-net in base 3, because
- 1 times m-reduction [i] would yield (170, 214, 198091)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 270441 261088 628191 388631 921742 421836 993626 359449 716794 069774 679227 281569 416698 614867 504519 139013 617997 > 3214 [i]