Best Known (67−30, 67, s)-Nets in Base 256
(67−30, 67, 4370)-Net over F256 — Constructive and digital
Digital (37, 67, 4370)-net over F256, using
- 2561 times duplication [i] based on digital (36, 66, 4370)-net over F256, using
- t-expansion [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
- t-expansion [i] based on digital (35, 66, 4370)-net over F256, using
(67−30, 67, 26871)-Net over F256 — Digital
Digital (37, 67, 26871)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25667, 26871, F256, 2, 30) (dual of [(26871, 2), 53675, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25667, 32781, F256, 2, 30) (dual of [(32781, 2), 65495, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25667, 65562, F256, 30) (dual of [65562, 65495, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(20) [i] based on
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2568, 26, F256, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,256)), using
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- Reed–Solomon code RS(248,256) [i]
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- construction X applied to Ce(29) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(25667, 65562, F256, 30) (dual of [65562, 65495, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(25667, 32781, F256, 2, 30) (dual of [(32781, 2), 65495, 31]-NRT-code), using
(67−30, 67, large)-Net in Base 256 — Upper bound on s
There is no (37, 67, large)-net in base 256, because
- 28 times m-reduction [i] would yield (37, 39, large)-net in base 256, but