Best Known (57, 74, s)-Nets in Base 8
(57, 74, 4097)-Net over F8 — Constructive and digital
Digital (57, 74, 4097)-net over F8, using
- 81 times duplication [i] based on digital (56, 73, 4097)-net over F8, using
- net defined by OOA [i] based on linear OOA(873, 4097, F8, 17, 17) (dual of [(4097, 17), 69576, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(873, 32777, F8, 17) (dual of [32777, 32704, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(871, 32768, F8, 17) (dual of [32768, 32697, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(873, 32777, F8, 17) (dual of [32777, 32704, 18]-code), using
- net defined by OOA [i] based on linear OOA(873, 4097, F8, 17, 17) (dual of [(4097, 17), 69576, 18]-NRT-code), using
(57, 74, 22779)-Net over F8 — Digital
Digital (57, 74, 22779)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(874, 22779, F8, 17) (dual of [22779, 22705, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 32779, F8, 17) (dual of [32779, 32705, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(861, 32769, F8, 13) (dual of [32769, 32708, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(83, 10, F8, 3) (dual of [10, 7, 4]-code or 10-arc in PG(2,8) or 10-cap in PG(2,8)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(874, 32779, F8, 17) (dual of [32779, 32705, 18]-code), using
(57, 74, large)-Net in Base 8 — Upper bound on s
There is no (57, 74, large)-net in base 8, because
- 15 times m-reduction [i] would yield (57, 59, large)-net in base 8, but