Best Known (159, 159+41, s)-Nets in Base 3
(159, 159+41, 688)-Net over F3 — Constructive and digital
Digital (159, 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
(159, 159+41, 2058)-Net over F3 — Digital
Digital (159, 200, 2058)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3200, 2058, F3, 41) (dual of [2058, 1858, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3200, 2223, F3, 41) (dual of [2223, 2023, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- 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(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-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(3200, 2223, F3, 41) (dual of [2223, 2023, 42]-code), using
(159, 159+41, 232057)-Net in Base 3 — Upper bound on s
There is no (159, 200, 232058)-net in base 3, because
- 1 times m-reduction [i] would yield (159, 199, 232058)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 88539 287026 030068 856972 170625 582343 155665 651143 518521 780686 482578 491135 771801 748749 507227 483977 > 3199 [i]