Best Known (232−35, 232, s)-Nets in Base 3
(232−35, 232, 3474)-Net over F3 — Constructive and digital
Digital (197, 232, 3474)-net over F3, using
- 31 times duplication [i] based on digital (196, 231, 3474)-net over F3, using
- net defined by OOA [i] based on linear OOA(3231, 3474, F3, 35, 35) (dual of [(3474, 35), 121359, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(3231, 59059, F3, 35) (dual of [59059, 58828, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3231, 59049, F3, 35) (dual of [59049, 58818, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(34) ⊂ Ce(33) [i] based on
- OOA 17-folding and stacking with additional row [i] based on linear OA(3231, 59059, F3, 35) (dual of [59059, 58828, 36]-code), using
- net defined by OOA [i] based on linear OOA(3231, 3474, F3, 35, 35) (dual of [(3474, 35), 121359, 36]-NRT-code), using
(232−35, 232, 19686)-Net over F3 — Digital
Digital (197, 232, 19686)-net over F3, using
- 31 times duplication [i] based on digital (196, 231, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 19686, F3, 3, 35) (dual of [(19686, 3), 58827, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3231, 59058, F3, 35) (dual of [59058, 58827, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, 59059, F3, 35) (dual of [59059, 58828, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3231, 59049, F3, 35) (dual of [59049, 58818, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3231, 59059, F3, 35) (dual of [59059, 58828, 36]-code), using
- OOA 3-folding [i] based on linear OA(3231, 59058, F3, 35) (dual of [59058, 58827, 36]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 19686, F3, 3, 35) (dual of [(19686, 3), 58827, 36]-NRT-code), using
(232−35, 232, large)-Net in Base 3 — Upper bound on s
There is no (197, 232, large)-net in base 3, because
- 33 times m-reduction [i] would yield (197, 199, large)-net in base 3, but