Best Known (72, 72+25, s)-Nets in Base 27
(72, 72+25, 44287)-Net over F27 — Constructive and digital
Digital (72, 97, 44287)-net over F27, using
- net defined by OOA [i] based on linear OOA(2797, 44287, F27, 25, 25) (dual of [(44287, 25), 1107078, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2797, 531445, F27, 25) (dual of [531445, 531348, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(2797, 531441, F27, 25) (dual of [531441, 531344, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2793, 531441, F27, 24) (dual of [531441, 531348, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(2797, 531445, F27, 25) (dual of [531445, 531348, 26]-code), using
(72, 72+25, 341866)-Net over F27 — Digital
Digital (72, 97, 341866)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2797, 341866, F27, 25) (dual of [341866, 341769, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2797, 531441, F27, 25) (dual of [531441, 531344, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(2797, 531441, F27, 25) (dual of [531441, 531344, 26]-code), using
(72, 72+25, large)-Net in Base 27 — Upper bound on s
There is no (72, 97, large)-net in base 27, because
- 23 times m-reduction [i] would yield (72, 74, large)-net in base 27, but