Best Known (137, 169, s)-Nets in Base 3
(137, 169, 688)-Net over F3 — Constructive and digital
Digital (137, 169, 688)-net over F3, using
- t-expansion [i] based on digital (136, 169, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 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, 43, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (136, 172, 688)-net over F3, using
(137, 169, 3235)-Net over F3 — Digital
Digital (137, 169, 3235)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3169, 3235, F3, 2, 32) (dual of [(3235, 2), 6301, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3169, 3284, F3, 2, 32) (dual of [(3284, 2), 6399, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3169, 6568, F3, 32) (dual of [6568, 6399, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 6569, F3, 32) (dual of [6569, 6400, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3169, 6569, F3, 32) (dual of [6569, 6400, 33]-code), using
- OOA 2-folding [i] based on linear OA(3169, 6568, F3, 32) (dual of [6568, 6399, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(3169, 3284, F3, 2, 32) (dual of [(3284, 2), 6399, 33]-NRT-code), using
(137, 169, 372463)-Net in Base 3 — Upper bound on s
There is no (137, 169, 372464)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 430 029592 214697 946907 103644 316774 928811 900629 791982 976505 199657 228169 670856 219649 > 3169 [i]