Best Known (159−13, 159, s)-Nets in Base 3
(159−13, 159, 1594328)-Net over F3 — Constructive and digital
Digital (146, 159, 1594328)-net over F3, using
- 31 times duplication [i] based on digital (145, 158, 1594328)-net over F3, using
- trace code for nets [i] based on digital (66, 79, 797164)-net over F9, using
- net defined by OOA [i] based on linear OOA(979, 797164, F9, 13, 13) (dual of [(797164, 13), 10363053, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(979, 4782985, F9, 13) (dual of [4782985, 4782906, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(915, 16, F9, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,9)), using
- dual of repetition code with length 16 [i]
- linear OA(91, 16, F9, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(979, 4782985, F9, 13) (dual of [4782985, 4782906, 14]-code), using
- net defined by OOA [i] based on linear OOA(979, 797164, F9, 13, 13) (dual of [(797164, 13), 10363053, 14]-NRT-code), using
- trace code for nets [i] based on digital (66, 79, 797164)-net over F9, using
(159−13, 159, large)-Net over F3 — Digital
Digital (146, 159, large)-net over F3, using
- 37 times duplication [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
(159−13, 159, large)-Net in Base 3 — Upper bound on s
There is no (146, 159, large)-net in base 3, because
- 11 times m-reduction [i] would yield (146, 148, large)-net in base 3, but