Best Known (101, 101+20, s)-Nets in Base 3
(101, 101+20, 1969)-Net over F3 — Constructive and digital
Digital (101, 121, 1969)-net over F3, using
- 33 times duplication [i] based on digital (98, 118, 1969)-net over F3, using
- net defined by OOA [i] based on linear OOA(3118, 1969, F3, 20, 20) (dual of [(1969, 20), 39262, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3118, 19690, F3, 20) (dual of [19690, 19572, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3118, 19692, F3, 20) (dual of [19692, 19574, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3118, 19692, F3, 20) (dual of [19692, 19574, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3118, 19690, F3, 20) (dual of [19690, 19572, 21]-code), using
- net defined by OOA [i] based on linear OOA(3118, 1969, F3, 20, 20) (dual of [(1969, 20), 39262, 21]-NRT-code), using
(101, 101+20, 7832)-Net over F3 — Digital
Digital (101, 121, 7832)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3121, 7832, F3, 2, 20) (dual of [(7832, 2), 15543, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3121, 9848, F3, 2, 20) (dual of [(9848, 2), 19575, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3121, 19696, F3, 20) (dual of [19696, 19575, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3121, 19696, F3, 20) (dual of [19696, 19575, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3121, 9848, F3, 2, 20) (dual of [(9848, 2), 19575, 21]-NRT-code), using
(101, 101+20, 1343106)-Net in Base 3 — Upper bound on s
There is no (101, 121, 1343107)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5391 055061 387790 819709 314252 860594 991677 429655 077082 899541 > 3121 [i]