Best Known (164−35, 164, s)-Nets in Base 3
(164−35, 164, 640)-Net over F3 — Constructive and digital
Digital (129, 164, 640)-net over F3, using
- t-expansion [i] based on digital (128, 164, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 41, 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, 41, 160)-net over F81, using
(164−35, 164, 1466)-Net over F3 — Digital
Digital (129, 164, 1466)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3164, 1466, F3, 35) (dual of [1466, 1302, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3164, 2197, F3, 35) (dual of [2197, 2033, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(33) ⊂ 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(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(34) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3164, 2197, F3, 35) (dual of [2197, 2033, 36]-code), using
(164−35, 164, 134776)-Net in Base 3 — Upper bound on s
There is no (129, 164, 134777)-net in base 3, because
- 1 times m-reduction [i] would yield (129, 163, 134777)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 589947 707901 407685 907412 484084 074451 185299 026632 511805 621113 559948 153228 235187 > 3163 [i]