Best Known (123, 123+20, s)-Nets in Base 3
(123, 123+20, 5913)-Net over F3 — Constructive and digital
Digital (123, 143, 5913)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 12, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (111, 131, 5905)-net over F3, using
- net defined by OOA [i] based on linear OOA(3131, 5905, F3, 20, 20) (dual of [(5905, 20), 117969, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3131, 59050, F3, 20) (dual of [59050, 58919, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3131, 59059, F3, 20) (dual of [59059, 58928, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3131, 59059, F3, 20) (dual of [59059, 58928, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3131, 59050, F3, 20) (dual of [59050, 58919, 21]-code), using
- net defined by OOA [i] based on linear OOA(3131, 5905, F3, 20, 20) (dual of [(5905, 20), 117969, 21]-NRT-code), using
- digital (2, 12, 8)-net over F3, using
(123, 123+20, 29550)-Net over F3 — Digital
Digital (123, 143, 29550)-net over F3, using
- 31 times duplication [i] based on digital (122, 142, 29550)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3142, 29550, F3, 2, 20) (dual of [(29550, 2), 58958, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3142, 59100, F3, 20) (dual of [59100, 58958, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3142, 59100, F3, 20) (dual of [59100, 58958, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3142, 29550, F3, 2, 20) (dual of [(29550, 2), 58958, 21]-NRT-code), using
(123, 123+20, large)-Net in Base 3 — Upper bound on s
There is no (123, 143, large)-net in base 3, because
- 18 times m-reduction [i] would yield (123, 125, large)-net in base 3, but