Best Known (199, 231, s)-Nets in Base 3
(199, 231, 3703)-Net over F3 — Constructive and digital
Digital (199, 231, 3703)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 20, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (179, 211, 3691)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 3691, F3, 32, 32) (dual of [(3691, 32), 117901, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3211, 59056, F3, 32) (dual of [59056, 58845, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 59059, F3, 32) (dual of [59059, 58848, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- 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(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 59059, F3, 32) (dual of [59059, 58848, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3211, 59056, F3, 32) (dual of [59056, 58845, 33]-code), using
- net defined by OOA [i] based on linear OOA(3211, 3691, F3, 32, 32) (dual of [(3691, 32), 117901, 33]-NRT-code), using
- digital (4, 20, 12)-net over F3, using
(199, 231, 29560)-Net over F3 — Digital
Digital (199, 231, 29560)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 29560, F3, 2, 32) (dual of [(29560, 2), 58889, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3231, 59120, F3, 32) (dual of [59120, 58889, 33]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(3226, 59114, F3, 32) (dual of [59114, 58888, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- 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(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(315, 65, F3, 6) (dual of [65, 50, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(31) ⊂ Ce(24) [i] based on
- linear OA(3226, 59115, F3, 27) (dual of [59115, 58889, 28]-code), using Gilbert–Varšamov bound and bm = 3226 > Vbs−1(k−1) = 1917 862275 617060 586840 663518 725615 888366 338598 140496 768996 758472 955172 027472 597059 535343 125552 349869 895721 [i]
- linear OA(34, 5, F3, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,3)), using
- dual of repetition code with length 5 [i]
- linear OA(3226, 59114, F3, 32) (dual of [59114, 58888, 33]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(3231, 59120, F3, 32) (dual of [59120, 58889, 33]-code), using
(199, 231, large)-Net in Base 3 — Upper bound on s
There is no (199, 231, large)-net in base 3, because
- 30 times m-reduction [i] would yield (199, 201, large)-net in base 3, but