Best Known (52−23, 52, s)-Nets in Base 256
(52−23, 52, 5959)-Net over F256 — Constructive and digital
Digital (29, 52, 5959)-net over F256, using
- 2563 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
(52−23, 52, 32780)-Net over F256 — Digital
Digital (29, 52, 32780)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25652, 32780, F256, 2, 23) (dual of [(32780, 2), 65508, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25652, 65560, F256, 23) (dual of [65560, 65508, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [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(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(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([0,11]) ⊂ C([0,7]) [i] based on
- OOA 2-folding [i] based on linear OA(25652, 65560, F256, 23) (dual of [65560, 65508, 24]-code), using
(52−23, 52, large)-Net in Base 256 — Upper bound on s
There is no (29, 52, large)-net in base 256, because
- 21 times m-reduction [i] would yield (29, 31, large)-net in base 256, but