Best Known (89, 106, s)-Nets in Base 3
(89, 106, 2464)-Net over F3 — Constructive and digital
Digital (89, 106, 2464)-net over F3, using
- net defined by OOA [i] based on linear OOA(3106, 2464, F3, 17, 17) (dual of [(2464, 17), 41782, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3106, 19713, F3, 17) (dual of [19713, 19607, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 19716, F3, 17) (dual of [19716, 19610, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 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(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3106, 19716, F3, 17) (dual of [19716, 19610, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3106, 19713, F3, 17) (dual of [19713, 19607, 18]-code), using
(89, 106, 9858)-Net over F3 — Digital
Digital (89, 106, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 9858, F3, 2, 17) (dual of [(9858, 2), 19610, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 19716, F3, 17) (dual of [19716, 19610, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 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(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(3106, 19716, F3, 17) (dual of [19716, 19610, 18]-code), using
(89, 106, 3442513)-Net in Base 3 — Upper bound on s
There is no (89, 106, 3442514)-net in base 3, because
- 1 times m-reduction [i] would yield (89, 105, 3442514)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 125 237022 048464 826908 876939 374544 792295 895788 136785 > 3105 [i]