Best Known (69, 83, s)-Nets in Base 3
(69, 83, 2813)-Net over F3 — Constructive and digital
Digital (69, 83, 2813)-net over F3, using
- 31 times duplication [i] based on digital (68, 82, 2813)-net over F3, using
- net defined by OOA [i] based on linear OOA(382, 2813, F3, 14, 14) (dual of [(2813, 14), 39300, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(382, 19691, F3, 14) (dual of [19691, 19609, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 19692, F3, 14) (dual of [19692, 19610, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(382, 19692, F3, 14) (dual of [19692, 19610, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(382, 19691, F3, 14) (dual of [19691, 19609, 15]-code), using
- net defined by OOA [i] based on linear OOA(382, 2813, F3, 14, 14) (dual of [(2813, 14), 39300, 15]-NRT-code), using
(69, 83, 7994)-Net over F3 — Digital
Digital (69, 83, 7994)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(383, 7994, F3, 2, 14) (dual of [(7994, 2), 15905, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(383, 9846, F3, 2, 14) (dual of [(9846, 2), 19609, 15]-NRT-code), using
- 31 times duplication [i] based on linear OOA(382, 9846, F3, 2, 14) (dual of [(9846, 2), 19610, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(382, 19692, F3, 14) (dual of [19692, 19610, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(382, 19692, F3, 14) (dual of [19692, 19610, 15]-code), using
- 31 times duplication [i] based on linear OOA(382, 9846, F3, 2, 14) (dual of [(9846, 2), 19610, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(383, 9846, F3, 2, 14) (dual of [(9846, 2), 19609, 15]-NRT-code), using
(69, 83, 767679)-Net in Base 3 — Upper bound on s
There is no (69, 83, 767680)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3990 862430 524274 621595 329524 458856 616193 > 383 [i]