Best Known (40−18, 40, s)-Nets in Base 256
(40−18, 40, 7283)-Net over F256 — Constructive and digital
Digital (22, 40, 7283)-net over F256, using
- t-expansion [i] based on digital (21, 40, 7283)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
(40−18, 40, 31770)-Net over F256 — Digital
Digital (22, 40, 31770)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25640, 31770, F256, 2, 18) (dual of [(31770, 2), 63500, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25640, 32776, F256, 2, 18) (dual of [(32776, 2), 65512, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25640, 65552, F256, 18) (dual of [65552, 65512, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, 65553, F256, 18) (dual of [65553, 65513, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- 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(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(25640, 65553, F256, 18) (dual of [65553, 65513, 19]-code), using
- OOA 2-folding [i] based on linear OA(25640, 65552, F256, 18) (dual of [65552, 65512, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(25640, 32776, F256, 2, 18) (dual of [(32776, 2), 65512, 19]-NRT-code), using
(40−18, 40, large)-Net in Base 256 — Upper bound on s
There is no (22, 40, large)-net in base 256, because
- 16 times m-reduction [i] would yield (22, 24, large)-net in base 256, but