Best Known (27, 27+21, s)-Nets in Base 256
(27, 27+21, 6555)-Net over F256 — Constructive and digital
Digital (27, 48, 6555)-net over F256, using
- 2562 times duplication [i] based on digital (25, 46, 6555)-net over F256, using
- net defined by OOA [i] based on linear OOA(25646, 6555, F256, 21, 21) (dual of [(6555, 21), 137609, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25646, 65551, F256, 21) (dual of [65551, 65505, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25646, 65554, F256, 21) (dual of [65554, 65508, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25646, 65554, F256, 21) (dual of [65554, 65508, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25646, 65551, F256, 21) (dual of [65551, 65505, 22]-code), using
- net defined by OOA [i] based on linear OOA(25646, 6555, F256, 21, 21) (dual of [(6555, 21), 137609, 22]-NRT-code), using
(27, 27+21, 32780)-Net over F256 — Digital
Digital (27, 48, 32780)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25648, 32780, F256, 2, 21) (dual of [(32780, 2), 65512, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25648, 65560, F256, 21) (dual of [65560, 65512, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2567, 23, F256, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- Reed–Solomon code RS(249,256) [i]
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(25648, 65560, F256, 21) (dual of [65560, 65512, 22]-code), using
(27, 27+21, large)-Net in Base 256 — Upper bound on s
There is no (27, 48, large)-net in base 256, because
- 19 times m-reduction [i] would yield (27, 29, large)-net in base 256, but