Best Known (221−33, 221, s)-Nets in Base 3
(221−33, 221, 3691)-Net over F3 — Constructive and digital
Digital (188, 221, 3691)-net over F3, using
- net defined by OOA [i] based on linear OOA(3221, 3691, F3, 33, 33) (dual of [(3691, 33), 121582, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3221, 59057, F3, 33) (dual of [59057, 58836, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 59059, F3, 33) (dual of [59059, 58838, 34]-code), using
- 1 times truncation [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- 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(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- 1 times truncation [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 59059, F3, 33) (dual of [59059, 58838, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3221, 59057, F3, 33) (dual of [59057, 58836, 34]-code), using
(221−33, 221, 19686)-Net over F3 — Digital
Digital (188, 221, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 19686, F3, 3, 33) (dual of [(19686, 3), 58837, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3221, 59058, F3, 33) (dual of [59058, 58837, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 59059, F3, 33) (dual of [59059, 58838, 34]-code), using
- 1 times truncation [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- 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(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- 1 times truncation [i] based on linear OA(3222, 59060, F3, 34) (dual of [59060, 58838, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 59059, F3, 33) (dual of [59059, 58838, 34]-code), using
- OOA 3-folding [i] based on linear OA(3221, 59058, F3, 33) (dual of [59058, 58837, 34]-code), using
(221−33, 221, large)-Net in Base 3 — Upper bound on s
There is no (188, 221, large)-net in base 3, because
- 31 times m-reduction [i] would yield (188, 190, large)-net in base 3, but