Best Known (114, 114+29, s)-Nets in Base 3
(114, 114+29, 640)-Net over F3 — Constructive and digital
Digital (114, 143, 640)-net over F3, using
- t-expansion [i] based on digital (113, 143, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (113, 144, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 36, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 36, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (113, 144, 640)-net over F3, using
(114, 114+29, 1740)-Net over F3 — Digital
Digital (114, 143, 1740)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3143, 1740, F3, 29) (dual of [1740, 1597, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3143, 2219, F3, 29) (dual of [2219, 2076, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(36, 29, F3, 3) (dual of [29, 23, 4]-code or 29-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), 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(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3143, 2219, F3, 29) (dual of [2219, 2076, 30]-code), using
(114, 114+29, 208820)-Net in Base 3 — Upper bound on s
There is no (114, 143, 208821)-net in base 3, because
- 1 times m-reduction [i] would yield (114, 142, 208821)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 56 395315 703433 107875 377109 730475 691910 878881 786894 710623 243860 342081 > 3142 [i]