Best Known (56, 75, s)-Nets in Base 8
(56, 75, 910)-Net over F8 — Constructive and digital
Digital (56, 75, 910)-net over F8, using
- 81 times duplication [i] based on digital (55, 74, 910)-net over F8, using
- net defined by OOA [i] based on linear OOA(874, 910, F8, 19, 19) (dual of [(910, 19), 17216, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(874, 8191, F8, 19) (dual of [8191, 8117, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 8194, F8, 19) (dual of [8194, 8120, 20]-code), using
- trace code [i] based on linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- trace code [i] based on linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 8194, F8, 19) (dual of [8194, 8120, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(874, 8191, F8, 19) (dual of [8191, 8117, 20]-code), using
- net defined by OOA [i] based on linear OOA(874, 910, F8, 19, 19) (dual of [(910, 19), 17216, 20]-NRT-code), using
(56, 75, 8198)-Net over F8 — Digital
Digital (56, 75, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(875, 8198, F8, 19) (dual of [8198, 8123, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(874, 8196, F8, 19) (dual of [8196, 8122, 20]-code), using
- trace code [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
- linear OA(874, 8197, F8, 18) (dual of [8197, 8123, 19]-code), using Gilbert–Varšamov bound and bm = 874 > Vbs−1(k−1) = 2 186497 457217 122394 473217 950324 453710 912907 841059 381144 778470 950400 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(874, 8196, F8, 19) (dual of [8196, 8122, 20]-code), using
- construction X with Varšamov bound [i] based on
(56, 75, large)-Net in Base 8 — Upper bound on s
There is no (56, 75, large)-net in base 8, because
- 17 times m-reduction [i] would yield (56, 58, large)-net in base 8, but