Best Known (165, 165+25, s)-Nets in Base 3
(165, 165+25, 14769)-Net over F3 — Constructive and digital
Digital (165, 190, 14769)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (152, 177, 14762)-net over F3, using
- net defined by OOA [i] based on linear OOA(3177, 14762, F3, 25, 25) (dual of [(14762, 25), 368873, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3177, 177145, F3, 25) (dual of [177145, 176968, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3177, 177145, F3, 25) (dual of [177145, 176968, 26]-code), using
- net defined by OOA [i] based on linear OOA(3177, 14762, F3, 25, 25) (dual of [(14762, 25), 368873, 26]-NRT-code), using
- digital (1, 13, 7)-net over F3, using
(165, 165+25, 59068)-Net over F3 — Digital
Digital (165, 190, 59068)-net over F3, using
- 31 times duplication [i] based on digital (164, 189, 59068)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3189, 59068, F3, 3, 25) (dual of [(59068, 3), 177015, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3189, 177204, F3, 25) (dual of [177204, 177015, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(312, 56, F3, 5) (dual of [56, 44, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 3-folding [i] based on linear OA(3189, 177204, F3, 25) (dual of [177204, 177015, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3189, 59068, F3, 3, 25) (dual of [(59068, 3), 177015, 26]-NRT-code), using
(165, 165+25, large)-Net in Base 3 — Upper bound on s
There is no (165, 190, large)-net in base 3, because
- 23 times m-reduction [i] would yield (165, 167, large)-net in base 3, but