Best Known (159, 159+42, s)-Nets in Base 3
(159, 159+42, 688)-Net over F3 — Constructive and digital
Digital (159, 201, 688)-net over F3, using
- 31 times duplication [i] based on digital (158, 200, 688)-net over F3, using
- t-expansion [i] based on digital (157, 200, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 50, 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, 50, 172)-net over F81, using
- t-expansion [i] based on digital (157, 200, 688)-net over F3, using
(159, 159+42, 1879)-Net over F3 — Digital
Digital (159, 201, 1879)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3201, 1879, F3, 42) (dual of [1879, 1678, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3201, 2207, F3, 42) (dual of [2207, 2006, 43]-code), using
- construction XX applied to Ce(42) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- 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(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [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(42) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3201, 2207, F3, 42) (dual of [2207, 2006, 43]-code), using
(159, 159+42, 160006)-Net in Base 3 — Upper bound on s
There is no (159, 201, 160007)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 796843 377201 327017 184408 971442 503282 545886 750510 971636 044077 978751 422732 130259 783624 517332 064295 > 3201 [i]