Best Known (165, 165+16, s)-Nets in Base 3
(165, 165+16, 1048739)-Net over F3 — Constructive and digital
Digital (165, 181, 1048739)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (22, 30, 164)-net over F3, using
- trace code for nets [i] based on digital (7, 15, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(7,81) in PG(14,9)) for nets [i] based on digital (0, 8, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base reduction for projective spaces (embedding PG(7,81) in PG(14,9)) for nets [i] based on digital (0, 8, 82)-net over F81, using
- trace code for nets [i] based on digital (7, 15, 82)-net over F9, using
- digital (135, 151, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- digital (22, 30, 164)-net over F3, using
(165, 165+16, 4194566)-Net over F3 — Digital
Digital (165, 181, 4194566)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 4194566, F3, 2, 16) (dual of [(4194566, 2), 8388951, 17]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(330, 265, F3, 2, 8) (dual of [(265, 2), 500, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(330, 265, F3, 8) (dual of [265, 235, 9]-code), using
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0) [i] based on linear OA(326, 248, F3, 8) (dual of [248, 222, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 243, F3, 7) (dual of [243, 222, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 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(7) ⊂ Ce(6) [i] based on
- 13 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0) [i] based on linear OA(326, 248, F3, 8) (dual of [248, 222, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(330, 265, F3, 8) (dual of [265, 235, 9]-code), using
- linear OOA(3151, 4194301, F3, 2, 16) (dual of [(4194301, 2), 8388451, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3151, 8388602, F3, 16) (dual of [8388602, 8388451, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OOA 2-folding [i] based on linear OA(3151, 8388602, F3, 16) (dual of [8388602, 8388451, 17]-code), using
- linear OOA(330, 265, F3, 2, 8) (dual of [(265, 2), 500, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
(165, 165+16, large)-Net in Base 3 — Upper bound on s
There is no (165, 181, large)-net in base 3, because
- 14 times m-reduction [i] would yield (165, 167, large)-net in base 3, but