Best Known (75, 75+25, s)-Nets in Base 27
(75, 75+25, 44288)-Net over F27 — Constructive and digital
Digital (75, 100, 44288)-net over F27, using
- net defined by OOA [i] based on linear OOA(27100, 44288, F27, 25, 25) (dual of [(44288, 25), 1107100, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(27100, 531457, F27, 25) (dual of [531457, 531357, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(27100, 531461, F27, 25) (dual of [531461, 531361, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2797, 531442, F27, 25) (dual of [531442, 531345, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2781, 531442, F27, 21) (dual of [531442, 531361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(273, 19, F27, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,27) or 19-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(27100, 531461, F27, 25) (dual of [531461, 531361, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(27100, 531457, F27, 25) (dual of [531457, 531357, 26]-code), using
(75, 75+25, 525486)-Net over F27 — Digital
Digital (75, 100, 525486)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27100, 525486, F27, 25) (dual of [525486, 525386, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(27100, 531461, F27, 25) (dual of [531461, 531361, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2797, 531442, F27, 25) (dual of [531442, 531345, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2781, 531442, F27, 21) (dual of [531442, 531361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(273, 19, F27, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,27) or 19-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(27100, 531461, F27, 25) (dual of [531461, 531361, 26]-code), using
(75, 75+25, large)-Net in Base 27 — Upper bound on s
There is no (75, 100, large)-net in base 27, because
- 23 times m-reduction [i] would yield (75, 77, large)-net in base 27, but