Best Known (123−20, 123, s)-Nets in Base 3
(123−20, 123, 1970)-Net over F3 — Constructive and digital
Digital (103, 123, 1970)-net over F3, using
- 31 times duplication [i] based on digital (102, 122, 1970)-net over F3, using
- net defined by OOA [i] based on linear OOA(3122, 1970, F3, 20, 20) (dual of [(1970, 20), 39278, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3122, 19700, F3, 20) (dual of [19700, 19578, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 19705, F3, 20) (dual of [19705, 19583, 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(34, 22, F3, 2) (dual of [22, 18, 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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3122, 19705, F3, 20) (dual of [19705, 19583, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3122, 19700, F3, 20) (dual of [19700, 19578, 21]-code), using
- net defined by OOA [i] based on linear OOA(3122, 1970, F3, 20, 20) (dual of [(1970, 20), 39278, 21]-NRT-code), using
(123−20, 123, 8915)-Net over F3 — Digital
Digital (103, 123, 8915)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3123, 8915, F3, 2, 20) (dual of [(8915, 2), 17707, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3123, 9853, F3, 2, 20) (dual of [(9853, 2), 19583, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3123, 19706, F3, 20) (dual of [19706, 19583, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3122, 19705, F3, 20) (dual of [19705, 19583, 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(34, 22, F3, 2) (dual of [22, 18, 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(19) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3122, 19705, F3, 20) (dual of [19705, 19583, 21]-code), using
- OOA 2-folding [i] based on linear OA(3123, 19706, F3, 20) (dual of [19706, 19583, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3123, 9853, F3, 2, 20) (dual of [(9853, 2), 19583, 21]-NRT-code), using
(123−20, 123, 1673151)-Net in Base 3 — Upper bound on s
There is no (103, 123, 1673152)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 48519 414487 992370 302343 909504 621790 114993 103273 948719 700609 > 3123 [i]