Best Known (125, 125+20, s)-Nets in Base 3
(125, 125+20, 17715)-Net over F3 — Constructive and digital
Digital (125, 145, 17715)-net over F3, using
- 31 times duplication [i] based on digital (124, 144, 17715)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 17715, F3, 20, 20) (dual of [(17715, 20), 354156, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3144, 177150, F3, 20) (dual of [177150, 177006, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 177158, F3, 20) (dual of [177158, 177014, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3144, 177158, F3, 20) (dual of [177158, 177014, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3144, 177150, F3, 20) (dual of [177150, 177006, 21]-code), using
- net defined by OOA [i] based on linear OOA(3144, 17715, F3, 20, 20) (dual of [(17715, 20), 354156, 21]-NRT-code), using
(125, 125+20, 58324)-Net over F3 — Digital
Digital (125, 145, 58324)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3145, 58324, F3, 3, 20) (dual of [(58324, 3), 174827, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3145, 59053, F3, 3, 20) (dual of [(59053, 3), 177014, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3145, 177159, F3, 20) (dual of [177159, 177014, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3144, 177158, F3, 20) (dual of [177158, 177014, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3144, 177158, F3, 20) (dual of [177158, 177014, 21]-code), using
- OOA 3-folding [i] based on linear OA(3145, 177159, F3, 20) (dual of [177159, 177014, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3145, 59053, F3, 3, 20) (dual of [(59053, 3), 177014, 21]-NRT-code), using
(125, 125+20, large)-Net in Base 3 — Upper bound on s
There is no (125, 145, large)-net in base 3, because
- 18 times m-reduction [i] would yield (125, 127, large)-net in base 3, but