Best Known (54−25, 54, s)-Nets in Base 256
(54−25, 54, 5462)-Net over F256 — Constructive and digital
Digital (29, 54, 5462)-net over F256, using
- 2562 times duplication [i] based on digital (27, 52, 5462)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 5462, F256, 25, 25) (dual of [(5462, 25), 136498, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25652, 65545, F256, 25) (dual of [65545, 65493, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25652, 65545, F256, 25) (dual of [65545, 65493, 26]-code), using
- net defined by OOA [i] based on linear OOA(25652, 5462, F256, 25, 25) (dual of [(5462, 25), 136498, 26]-NRT-code), using
(54−25, 54, 21851)-Net over F256 — Digital
Digital (29, 54, 21851)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25654, 21851, F256, 3, 25) (dual of [(21851, 3), 65499, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25654, 65553, F256, 25) (dual of [65553, 65499, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25654, 65554, F256, 25) (dual of [65554, 65500, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25654, 65554, F256, 25) (dual of [65554, 65500, 26]-code), using
- OOA 3-folding [i] based on linear OA(25654, 65553, F256, 25) (dual of [65553, 65499, 26]-code), using
(54−25, 54, large)-Net in Base 256 — Upper bound on s
There is no (29, 54, large)-net in base 256, because
- 23 times m-reduction [i] would yield (29, 31, large)-net in base 256, but