Best Known (133, 133+33, s)-Nets in Base 3
(133, 133+33, 688)-Net over F3 — Constructive and digital
Digital (133, 166, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(133, 133+33, 2121)-Net over F3 — Digital
Digital (133, 166, 2121)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3166, 2121, F3, 33) (dual of [2121, 1955, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 2227, F3, 33) (dual of [2227, 2061, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(3155, 2188, F3, 33) (dual of [2188, 2033, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(311, 39, F3, 5) (dual of [39, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(37, 14, F3, 5) (dual of [14, 7, 6]-code), using
- extended quadratic residue code Qe(14,3) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 2227, F3, 33) (dual of [2227, 2061, 34]-code), using
(133, 133+33, 283007)-Net in Base 3 — Upper bound on s
There is no (133, 166, 283008)-net in base 3, because
- 1 times m-reduction [i] would yield (133, 165, 283008)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5 309092 421979 854517 412896 717750 841803 748753 336449 158509 797518 697423 441051 754497 > 3165 [i]