Best Known (134, 134+31, s)-Nets in Base 3
(134, 134+31, 688)-Net over F3 — Constructive and digital
Digital (134, 165, 688)-net over F3, using
- t-expansion [i] based on digital (133, 165, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
(134, 134+31, 3290)-Net over F3 — Digital
Digital (134, 165, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3165, 3290, F3, 2, 31) (dual of [(3290, 2), 6415, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3165, 6580, F3, 31) (dual of [6580, 6415, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3165, 6581, F3, 31) (dual of [6581, 6416, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- 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(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3165, 6581, F3, 31) (dual of [6581, 6416, 32]-code), using
- OOA 2-folding [i] based on linear OA(3165, 6580, F3, 31) (dual of [6580, 6415, 32]-code), using
(134, 134+31, 528750)-Net in Base 3 — Upper bound on s
There is no (134, 165, 528751)-net in base 3, because
- 1 times m-reduction [i] would yield (134, 164, 528751)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 769689 161697 926277 541164 795953 604789 747186 790447 667297 983499 927794 688792 767907 > 3164 [i]