Best Known (156−27, 156, s)-Nets in Base 3
(156−27, 156, 704)-Net over F3 — Constructive and digital
Digital (129, 156, 704)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
- digital (7, 20, 16)-net over F3, using
(156−27, 156, 4597)-Net over F3 — Digital
Digital (129, 156, 4597)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3156, 4597, F3, 27) (dual of [4597, 4441, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3156, 6603, F3, 27) (dual of [6603, 6447, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [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 C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3156, 6603, F3, 27) (dual of [6603, 6447, 28]-code), using
(156−27, 156, 1383966)-Net in Base 3 — Upper bound on s
There is no (129, 156, 1383967)-net in base 3, because
- 1 times m-reduction [i] would yield (129, 155, 1383967)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 89 907344 636421 315425 928857 658887 938005 742001 314423 610620 383102 043939 953447 > 3155 [i]