Best Known (189, 189+31, s)-Nets in Base 3
(189, 189+31, 3948)-Net over F3 — Constructive and digital
Digital (189, 220, 3948)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 19, 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 (170, 201, 3936)-net over F3, using
- net defined by OOA [i] based on linear OOA(3201, 3936, F3, 31, 31) (dual of [(3936, 31), 121815, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3201, 59041, F3, 31) (dual of [59041, 58840, 32]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3201, 59041, F3, 31) (dual of [59041, 58840, 32]-code), using
- net defined by OOA [i] based on linear OOA(3201, 3936, F3, 31, 31) (dual of [(3936, 31), 121815, 32]-NRT-code), using
- digital (4, 19, 12)-net over F3, using
(189, 189+31, 29262)-Net over F3 — Digital
Digital (189, 220, 29262)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3220, 29262, F3, 2, 31) (dual of [(29262, 2), 58304, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3220, 29559, F3, 2, 31) (dual of [(29559, 2), 58898, 32]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3218, 29558, F3, 2, 31) (dual of [(29558, 2), 58898, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3218, 59116, F3, 31) (dual of [59116, 58898, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- 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(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(317, 67, F3, 7) (dual of [67, 50, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(3218, 59116, F3, 31) (dual of [59116, 58898, 32]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3218, 29558, F3, 2, 31) (dual of [(29558, 2), 58898, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3220, 29559, F3, 2, 31) (dual of [(29559, 2), 58898, 32]-NRT-code), using
(189, 189+31, large)-Net in Base 3 — Upper bound on s
There is no (189, 220, large)-net in base 3, because
- 29 times m-reduction [i] would yield (189, 191, large)-net in base 3, but