Best Known (63, 63+20, s)-Nets in Base 27
(63, 63+20, 53147)-Net over F27 — Constructive and digital
Digital (63, 83, 53147)-net over F27, using
- net defined by OOA [i] based on linear OOA(2783, 53147, F27, 20, 20) (dual of [(53147, 20), 1062857, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2783, 531470, F27, 20) (dual of [531470, 531387, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2783, 531471, F27, 20) (dual of [531471, 531388, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(2777, 531441, F27, 20) (dual of [531441, 531364, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2753, 531441, F27, 14) (dual of [531441, 531388, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2783, 531471, F27, 20) (dual of [531471, 531388, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2783, 531470, F27, 20) (dual of [531470, 531387, 21]-code), using
(63, 63+20, 545820)-Net over F27 — Digital
Digital (63, 83, 545820)-net over F27, using
(63, 63+20, large)-Net in Base 27 — Upper bound on s
There is no (63, 83, large)-net in base 27, because
- 18 times m-reduction [i] would yield (63, 65, large)-net in base 27, but