Best Known (37, 68, s)-Nets in Base 256
(37, 68, 4370)-Net over F256 — Constructive and digital
Digital (37, 68, 4370)-net over F256, using
- 2562 times duplication [i] based on digital (35, 66, 4370)-net over F256, using
- net defined by OOA [i] based on linear OOA(25666, 4370, F256, 31, 31) (dual of [(4370, 31), 135404, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25666, 65551, F256, 31) (dual of [65551, 65485, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 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(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([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25666, 65551, F256, 31) (dual of [65551, 65485, 32]-code), using
- net defined by OOA [i] based on linear OOA(25666, 4370, F256, 31, 31) (dual of [(4370, 31), 135404, 32]-NRT-code), using
(37, 68, 21853)-Net over F256 — Digital
Digital (37, 68, 21853)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25668, 21853, F256, 3, 31) (dual of [(21853, 3), 65491, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25668, 65559, F256, 31) (dual of [65559, 65491, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(25668, 65560, F256, 31) (dual of [65560, 65492, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,11]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 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(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([0,15]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25668, 65560, F256, 31) (dual of [65560, 65492, 32]-code), using
- OOA 3-folding [i] based on linear OA(25668, 65559, F256, 31) (dual of [65559, 65491, 32]-code), using
(37, 68, large)-Net in Base 256 — Upper bound on s
There is no (37, 68, large)-net in base 256, because
- 29 times m-reduction [i] would yield (37, 39, large)-net in base 256, but