Best Known (157−12, 157, s)-Nets in Base 3
(157−12, 157, 1594370)-Net over F3 — Constructive and digital
Digital (145, 157, 1594370)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 42)-net over F3, using
- digital (132, 144, 1594328)-net over F3, using
- trace code for nets [i] based on digital (60, 72, 797164)-net over F9, using
- net defined by OOA [i] based on linear OOA(972, 797164, F9, 12, 12) (dual of [(797164, 12), 9565896, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(972, 4782984, F9, 12) (dual of [4782984, 4782912, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(972, 4782984, F9, 12) (dual of [4782984, 4782912, 13]-code), using
- net defined by OOA [i] based on linear OOA(972, 797164, F9, 12, 12) (dual of [(797164, 12), 9565896, 13]-NRT-code), using
- trace code for nets [i] based on digital (60, 72, 797164)-net over F9, using
(157−12, 157, large)-Net over F3 — Digital
Digital (145, 157, large)-net over F3, using
- 35 times duplication [i] based on digital (140, 152, large)-net over F3, using
- t-expansion [i] based on digital (139, 152, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3152, large, F3, 13) (dual of [large, large−152, 14]-code), using
- 31 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 31 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3152, large, F3, 13) (dual of [large, large−152, 14]-code), using
- t-expansion [i] based on digital (139, 152, large)-net over F3, using
(157−12, 157, large)-Net in Base 3 — Upper bound on s
There is no (145, 157, large)-net in base 3, because
- 10 times m-reduction [i] would yield (145, 147, large)-net in base 3, but