Best Known (167−33, 167, s)-Nets in Base 3
(167−33, 167, 688)-Net over F3 — Constructive and digital
Digital (134, 167, 688)-net over F3, using
- t-expansion [i] based on digital (133, 167, 688)-net over F3, using
- 1 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
- 1 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
(167−33, 167, 2199)-Net over F3 — Digital
Digital (134, 167, 2199)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3167, 2199, F3, 33) (dual of [2199, 2032, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3167, 2228, F3, 33) (dual of [2228, 2061, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3165, 2226, F3, 33) (dual of [2226, 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(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3165, 2226, F3, 33) (dual of [2226, 2061, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3167, 2228, F3, 33) (dual of [2228, 2061, 34]-code), using
(167−33, 167, 303123)-Net in Base 3 — Upper bound on s
There is no (134, 167, 303124)-net in base 3, because
- 1 times m-reduction [i] would yield (134, 166, 303124)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 15 927250 419886 610159 622669 088007 951166 196383 965558 820300 436724 936066 830261 539969 > 3166 [i]