Best Known (227−24, 227, s)-Nets in Base 3
(227−24, 227, 398583)-Net over F3 — Constructive and digital
Digital (203, 227, 398583)-net over F3, using
- 31 times duplication [i] based on digital (202, 226, 398583)-net over F3, using
- net defined by OOA [i] based on linear OOA(3226, 398583, F3, 24, 24) (dual of [(398583, 24), 9565766, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3226, 4782996, F3, 24) (dual of [4782996, 4782770, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 4782998, F3, 24) (dual of [4782998, 4782772, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3225, 4782969, F3, 25) (dual of [4782969, 4782744, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(31, 29, F3, 1) (dual of [29, 28, 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(3226, 4782998, F3, 24) (dual of [4782998, 4782772, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3226, 4782996, F3, 24) (dual of [4782996, 4782770, 25]-code), using
- net defined by OOA [i] based on linear OOA(3226, 398583, F3, 24, 24) (dual of [(398583, 24), 9565766, 25]-NRT-code), using
(227−24, 227, 1195750)-Net over F3 — Digital
Digital (203, 227, 1195750)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3227, 1195750, F3, 4, 24) (dual of [(1195750, 4), 4782773, 25]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3227, 4783000, F3, 24) (dual of [4783000, 4782773, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3226, 4782999, F3, 24) (dual of [4782999, 4782773, 25]-code), using
- construction X4 applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3225, 4782969, F3, 25) (dual of [4782969, 4782744, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 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(3226, 4782999, F3, 24) (dual of [4782999, 4782773, 25]-code), using
- OOA 4-folding [i] based on linear OA(3227, 4783000, F3, 24) (dual of [4783000, 4782773, 25]-code), using
(227−24, 227, large)-Net in Base 3 — Upper bound on s
There is no (203, 227, large)-net in base 3, because
- 22 times m-reduction [i] would yield (203, 205, large)-net in base 3, but