Best Known (187, 187+24, s)-Nets in Base 3
(187, 187+24, 132862)-Net over F3 — Constructive and digital
Digital (187, 211, 132862)-net over F3, using
- 31 times duplication [i] based on digital (186, 210, 132862)-net over F3, using
- net defined by OOA [i] based on linear OOA(3210, 132862, F3, 24, 24) (dual of [(132862, 24), 3188478, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3210, 1594344, F3, 24) (dual of [1594344, 1594134, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 1594350, F3, 24) (dual of [1594350, 1594140, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(31, 27, F3, 1) (dual of [27, 26, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 1594350, F3, 24) (dual of [1594350, 1594140, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3210, 1594344, F3, 24) (dual of [1594344, 1594134, 25]-code), using
- net defined by OOA [i] based on linear OOA(3210, 132862, F3, 24, 24) (dual of [(132862, 24), 3188478, 25]-NRT-code), using
(187, 187+24, 398588)-Net over F3 — Digital
Digital (187, 211, 398588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3211, 398588, F3, 4, 24) (dual of [(398588, 4), 1594141, 25]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3211, 1594352, F3, 24) (dual of [1594352, 1594141, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3210, 1594351, F3, 24) (dual of [1594351, 1594141, 25]-code), using
- construction X4 applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(24) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3210, 1594351, F3, 24) (dual of [1594351, 1594141, 25]-code), using
- OOA 4-folding [i] based on linear OA(3211, 1594352, F3, 24) (dual of [1594352, 1594141, 25]-code), using
(187, 187+24, large)-Net in Base 3 — Upper bound on s
There is no (187, 211, large)-net in base 3, because
- 22 times m-reduction [i] would yield (187, 189, large)-net in base 3, but