Best Known (227−25, 227, s)-Nets in Base 3
(227−25, 227, 398582)-Net over F3 — Constructive and digital
Digital (202, 227, 398582)-net over F3, using
- net defined by OOA [i] based on linear OOA(3227, 398582, F3, 25, 25) (dual of [(398582, 25), 9964323, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3227, 4782985, F3, 25) (dual of [4782985, 4782758, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3226, 4782984, F3, 25) (dual of [4782984, 4782758, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [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(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3226, 4782984, F3, 25) (dual of [4782984, 4782758, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3227, 4782985, F3, 25) (dual of [4782985, 4782758, 26]-code), using
(227−25, 227, 956597)-Net over F3 — Digital
Digital (202, 227, 956597)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3227, 956597, F3, 5, 25) (dual of [(956597, 5), 4782758, 26]-NRT-code), using
- OOA 5-folding [i] based on linear OA(3227, 4782985, F3, 25) (dual of [4782985, 4782758, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3226, 4782984, F3, 25) (dual of [4782984, 4782758, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [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(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3226, 4782984, F3, 25) (dual of [4782984, 4782758, 26]-code), using
- OOA 5-folding [i] based on linear OA(3227, 4782985, F3, 25) (dual of [4782985, 4782758, 26]-code), using
(227−25, 227, large)-Net in Base 3 — Upper bound on s
There is no (202, 227, large)-net in base 3, because
- 23 times m-reduction [i] would yield (202, 204, large)-net in base 3, but