Best Known (174−29, 174, s)-Nets in Base 3
(174−29, 174, 1406)-Net over F3 — Constructive and digital
Digital (145, 174, 1406)-net over F3, using
- 32 times duplication [i] based on digital (143, 172, 1406)-net over F3, using
- net defined by OOA [i] based on linear OOA(3172, 1406, F3, 29, 29) (dual of [(1406, 29), 40602, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3172, 19685, F3, 29) (dual of [19685, 19513, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3172, 19692, F3, 29) (dual of [19692, 19520, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3172, 19692, F3, 29) (dual of [19692, 19520, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3172, 19685, F3, 29) (dual of [19685, 19513, 30]-code), using
- net defined by OOA [i] based on linear OOA(3172, 1406, F3, 29, 29) (dual of [(1406, 29), 40602, 30]-NRT-code), using
(174−29, 174, 7536)-Net over F3 — Digital
Digital (145, 174, 7536)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3174, 7536, F3, 2, 29) (dual of [(7536, 2), 14898, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3174, 9847, F3, 2, 29) (dual of [(9847, 2), 19520, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3172, 9846, F3, 2, 29) (dual of [(9846, 2), 19520, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3172, 19692, F3, 29) (dual of [19692, 19520, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3172, 19692, F3, 29) (dual of [19692, 19520, 30]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3172, 9846, F3, 2, 29) (dual of [(9846, 2), 19520, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3174, 9847, F3, 2, 29) (dual of [(9847, 2), 19520, 30]-NRT-code), using
(174−29, 174, 2378377)-Net in Base 3 — Upper bound on s
There is no (145, 174, 2378378)-net in base 3, because
- 1 times m-reduction [i] would yield (145, 173, 2378378)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 34831 999425 290291 894551 929743 347944 633624 454406 671745 712770 763396 161306 238126 283581 > 3173 [i]