Best Known (147−12, 147, s)-Nets in Base 3
(147−12, 147, 1594328)-Net over F3 — Constructive and digital
Digital (135, 147, 1594328)-net over F3, using
- 31 times duplication [i] based on digital (134, 146, 1594328)-net over F3, using
- net defined by OOA [i] based on linear OOA(3146, 1594328, F3, 14, 12) (dual of [(1594328, 14), 22320446, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3146, 4782985, F3, 2, 12) (dual of [(4782985, 2), 9565824, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 4782986, F3, 2, 12) (dual of [(4782986, 2), 9565826, 13]-NRT-code), using
- trace code [i] based on linear OOA(973, 2391493, F9, 2, 12) (dual of [(2391493, 2), 4782913, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(973, 4782986, F9, 12) (dual of [4782986, 4782913, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(972, 4782985, F9, 12) (dual of [4782985, 4782913, 13]-code), using
- construction X4 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(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(11) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(972, 4782985, F9, 12) (dual of [4782985, 4782913, 13]-code), using
- OOA 2-folding [i] based on linear OA(973, 4782986, F9, 12) (dual of [4782986, 4782913, 13]-code), using
- trace code [i] based on linear OOA(973, 2391493, F9, 2, 12) (dual of [(2391493, 2), 4782913, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 4782986, F3, 2, 12) (dual of [(4782986, 2), 9565826, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3146, 4782985, F3, 2, 12) (dual of [(4782985, 2), 9565824, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3146, 1594328, F3, 14, 12) (dual of [(1594328, 14), 22320446, 13]-NRT-code), using
(147−12, 147, large)-Net over F3 — Digital
Digital (135, 147, large)-net over F3, using
- 38 times duplication [i] based on digital (127, 139, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 19 times code embedding in larger space [i] based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3139, large, F3, 12) (dual of [large, large−139, 13]-code), using
(147−12, 147, large)-Net in Base 3 — Upper bound on s
There is no (135, 147, large)-net in base 3, because
- 10 times m-reduction [i] would yield (135, 137, large)-net in base 3, but