Best Known (75, 75+24, s)-Nets in Base 27
(75, 75+24, 44289)-Net over F27 — Constructive and digital
Digital (75, 99, 44289)-net over F27, using
- 271 times duplication [i] based on digital (74, 98, 44289)-net over F27, using
- net defined by OOA [i] based on linear OOA(2798, 44289, F27, 24, 24) (dual of [(44289, 24), 1062838, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2798, 531468, F27, 24) (dual of [531468, 531370, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2798, 531469, F27, 24) (dual of [531469, 531371, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- 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(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(275, 28, F27, 5) (dual of [28, 23, 6]-code or 28-arc in PG(4,27)), using
- extended Reed–Solomon code RSe(23,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(2798, 531469, F27, 24) (dual of [531469, 531371, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2798, 531468, F27, 24) (dual of [531468, 531370, 25]-code), using
- net defined by OOA [i] based on linear OOA(2798, 44289, F27, 24, 24) (dual of [(44289, 24), 1062838, 25]-NRT-code), using
(75, 75+24, 531471)-Net over F27 — Digital
Digital (75, 99, 531471)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2799, 531471, F27, 24) (dual of [531471, 531372, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
- 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(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(276, 30, F27, 5) (dual of [30, 24, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([1,2]) [i] based on
- linear OA(275, 28, F27, 5) (dual of [28, 23, 6]-code or 28-arc in PG(4,27)), using the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(274, 28, F27, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,27)), using the narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [1,2], and minimum distance d ≥ |{1,2}| + |{−3,0}| = 4 (simple Roos-bound) [i]
- linear OA(271, 2, F27, 1) (dual of [2, 1, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([1,2]) [i] based on
- construction X applied to Ce(23) ⊂ Ce(17) [i] based on
(75, 75+24, large)-Net in Base 27 — Upper bound on s
There is no (75, 99, large)-net in base 27, because
- 22 times m-reduction [i] would yield (75, 77, large)-net in base 27, but