Best Known (50−23, 50, s)-Nets in Base 256
(50−23, 50, 5959)-Net over F256 — Constructive and digital
Digital (27, 50, 5959)-net over F256, using
- 2561 times duplication [i] based on digital (26, 49, 5959)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5959, F256, 23, 23) (dual of [(5959, 23), 137008, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(25649, 65550, F256, 23) (dual of [65550, 65501, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(25649, 65550, F256, 23) (dual of [65550, 65501, 24]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5959, F256, 23, 23) (dual of [(5959, 23), 137008, 24]-NRT-code), using
(50−23, 50, 21851)-Net over F256 — Digital
Digital (27, 50, 21851)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25650, 21851, F256, 3, 23) (dual of [(21851, 3), 65503, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25650, 65553, F256, 23) (dual of [65553, 65503, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25650, 65554, F256, 23) (dual of [65554, 65504, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25650, 65554, F256, 23) (dual of [65554, 65504, 24]-code), using
- OOA 3-folding [i] based on linear OA(25650, 65553, F256, 23) (dual of [65553, 65503, 24]-code), using
(50−23, 50, large)-Net in Base 256 — Upper bound on s
There is no (27, 50, large)-net in base 256, because
- 21 times m-reduction [i] would yield (27, 29, large)-net in base 256, but