Best Known (155−29, 155, s)-Nets in Base 3
(155−29, 155, 688)-Net over F3 — Constructive and digital
Digital (126, 155, 688)-net over F3, using
- t-expansion [i] based on digital (124, 155, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (124, 156, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (124, 156, 688)-net over F3, using
(155−29, 155, 3286)-Net over F3 — Digital
Digital (126, 155, 3286)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3155, 3286, F3, 2, 29) (dual of [(3286, 2), 6417, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3155, 6572, F3, 29) (dual of [6572, 6417, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 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(28) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3155, 6572, F3, 29) (dual of [6572, 6417, 30]-code), using
(155−29, 155, 535489)-Net in Base 3 — Upper bound on s
There is no (126, 155, 535490)-net in base 3, because
- 1 times m-reduction [i] would yield (126, 154, 535490)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29 969809 140264 459174 194494 812890 690213 518728 654252 154395 183130 510530 285837 > 3154 [i]