Best Known (131, 131+35, s)-Nets in Base 3
(131, 131+35, 640)-Net over F3 — Constructive and digital
Digital (131, 166, 640)-net over F3, using
- 2 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
(131, 131+35, 1569)-Net over F3 — Digital
Digital (131, 166, 1569)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3166, 1569, F3, 35) (dual of [1569, 1403, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 2205, F3, 35) (dual of [2205, 2039, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(31) [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(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(34, 18, F3, 2) (dual of [18, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(34) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 2205, F3, 35) (dual of [2205, 2039, 36]-code), using
(131, 131+35, 153373)-Net in Base 3 — Upper bound on s
There is no (131, 166, 153374)-net in base 3, because
- 1 times m-reduction [i] would yield (131, 165, 153374)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5 309030 838541 185840 651628 131843 601165 717194 345977 547897 980953 265707 755631 679517 > 3165 [i]