Best Known (110−29, 110, s)-Nets in Base 27
(110−29, 110, 37960)-Net over F27 — Constructive and digital
Digital (81, 110, 37960)-net over F27, using
- 271 times duplication [i] based on digital (80, 109, 37960)-net over F27, using
- net defined by OOA [i] based on linear OOA(27109, 37960, F27, 29, 29) (dual of [(37960, 29), 1100731, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(27109, 531441, F27, 29) (dual of [531441, 531332, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 14-folding and stacking with additional row [i] based on linear OA(27109, 531441, F27, 29) (dual of [531441, 531332, 30]-code), using
- net defined by OOA [i] based on linear OOA(27109, 37960, F27, 29, 29) (dual of [(37960, 29), 1100731, 30]-NRT-code), using
(110−29, 110, 265723)-Net over F27 — Digital
Digital (81, 110, 265723)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(27110, 265723, F27, 2, 29) (dual of [(265723, 2), 531336, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(27110, 531446, F27, 29) (dual of [531446, 531336, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(27109, 531445, F27, 29) (dual of [531445, 531336, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(27109, 531441, F27, 29) (dual of [531441, 531332, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(27105, 531441, F27, 28) (dual of [531441, 531336, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(27109, 531445, F27, 29) (dual of [531445, 531336, 30]-code), using
- OOA 2-folding [i] based on linear OA(27110, 531446, F27, 29) (dual of [531446, 531336, 30]-code), using
(110−29, 110, large)-Net in Base 27 — Upper bound on s
There is no (81, 110, large)-net in base 27, because
- 27 times m-reduction [i] would yield (81, 83, large)-net in base 27, but