Best Known (204−23, 204, s)-Nets in Base 3
(204−23, 204, 144942)-Net over F3 — Constructive and digital
Digital (181, 204, 144942)-net over F3, using
- 32 times duplication [i] based on digital (179, 202, 144942)-net over F3, using
- net defined by OOA [i] based on linear OOA(3202, 144942, F3, 23, 23) (dual of [(144942, 23), 3333464, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3202, 1594363, F3, 23) (dual of [1594363, 1594161, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 1594368, F3, 23) (dual of [1594368, 1594166, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 1594368, F3, 23) (dual of [1594368, 1594166, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3202, 1594363, F3, 23) (dual of [1594363, 1594161, 24]-code), using
- net defined by OOA [i] based on linear OOA(3202, 144942, F3, 23, 23) (dual of [(144942, 23), 3333464, 24]-NRT-code), using
(204−23, 204, 442190)-Net over F3 — Digital
Digital (181, 204, 442190)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3204, 442190, F3, 3, 23) (dual of [(442190, 3), 1326366, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3204, 531456, F3, 3, 23) (dual of [(531456, 3), 1594164, 24]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3202, 531456, F3, 3, 23) (dual of [(531456, 3), 1594166, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3202, 1594368, F3, 23) (dual of [1594368, 1594166, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-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(22) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(3202, 1594368, F3, 23) (dual of [1594368, 1594166, 24]-code), using
- 32 times duplication [i] based on linear OOA(3202, 531456, F3, 3, 23) (dual of [(531456, 3), 1594166, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3204, 531456, F3, 3, 23) (dual of [(531456, 3), 1594164, 24]-NRT-code), using
(204−23, 204, large)-Net in Base 3 — Upper bound on s
There is no (181, 204, large)-net in base 3, because
- 21 times m-reduction [i] would yield (181, 183, large)-net in base 3, but