Best Known (132, 132+35, s)-Nets in Base 3
(132, 132+35, 640)-Net over F3 — Constructive and digital
Digital (132, 167, 640)-net over F3, using
- t-expansion [i] based on digital (131, 167, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (131, 168, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 42, 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, 42, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (131, 168, 640)-net over F3, using
(132, 132+35, 1623)-Net over F3 — Digital
Digital (132, 167, 1623)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3167, 1623, F3, 35) (dual of [1623, 1456, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3167, 2207, F3, 35) (dual of [2207, 2040, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- 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(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3167, 2207, F3, 35) (dual of [2207, 2040, 36]-code), using
(132, 132+35, 163613)-Net in Base 3 — Upper bound on s
There is no (132, 167, 163614)-net in base 3, because
- 1 times m-reduction [i] would yield (132, 166, 163614)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 15 926991 794014 142474 295600 391695 341950 355117 155456 032182 967762 355314 745244 674077 > 3166 [i]