Best Known (18, 18+5, s)-Nets in Base 8
(18, 18+5, 32264)-Net over F8 — Constructive and digital
Digital (18, 23, 32264)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 4033)-net over F8, using
- s-reduction based on digital (0, 0, s)-net over F8 with arbitrarily large s, using
- digital (0, 0, 4033)-net over F8 (see above)
- digital (0, 0, 4033)-net over F8 (see above)
- digital (0, 1, 4033)-net over F8, using
- s-reduction based on digital (0, 1, s)-net over F8 with arbitrarily large s, using
- digital (0, 1, 4033)-net over F8 (see above)
- digital (0, 1, 4033)-net over F8 (see above)
- digital (3, 5, 4033)-net over F8, using
- s-reduction based on digital (3, 5, 4681)-net over F8, using
- digital (10, 15, 4033)-net over F8, using
- net defined by OOA [i] based on linear OOA(815, 4033, F8, 5, 5) (dual of [(4033, 5), 20150, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(815, 8067, F8, 5) (dual of [8067, 8052, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(814, 8066, F8, 5) (dual of [8066, 8052, 6]-code), using
- trace code [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(814, 8066, F8, 5) (dual of [8066, 8052, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(815, 8067, F8, 5) (dual of [8067, 8052, 6]-code), using
- net defined by OOA [i] based on linear OOA(815, 4033, F8, 5, 5) (dual of [(4033, 5), 20150, 6]-NRT-code), using
- digital (0, 0, 4033)-net over F8, using
(18, 18+5, 64528)-Net over F8 — Digital
Digital (18, 23, 64528)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(823, 64528, F8, 5) (dual of [64528, 64505, 6]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(80, 8066, F8, 0) (dual of [8066, 8066, 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(80, 8066, F8, 0) (dual of [8066, 8066, 1]-code) (see above)
- linear OA(80, 8066, F8, 0) (dual of [8066, 8066, 1]-code) (see above)
- linear OA(81, 8066, F8, 1) (dual of [8066, 8065, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(81, 8066, F8, 1) (dual of [8066, 8065, 2]-code) (see above)
- linear OA(81, 8066, F8, 1) (dual of [8066, 8065, 2]-code) (see above)
- linear OA(86, 8066, F8, 2) (dual of [8066, 8060, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 32768, F8, 2) (dual of [32768, 32762, 3]-code), using
- an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(86, 32768, F8, 2) (dual of [32768, 32762, 3]-code), using
- linear OA(814, 8066, F8, 5) (dual of [8066, 8052, 6]-code), using
- trace code [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- linear OA(80, 8066, F8, 0) (dual of [8066, 8066, 1]-code), using
- generalized (u, u+v)-construction [i] based on
(18, 18+5, 65865)-Net in Base 8 — Constructive
(18, 23, 65865)-net in base 8, using
- net defined by OOA [i] based on OOA(823, 65865, S8, 6, 5), using
- OOA stacking with additional row [i] based on OOA(823, 65866, S8, 2, 5), using
- (u, u+v)-construction [i] based on
- linear OOA(84, 585, F8, 2, 2) (dual of [(585, 2), 1166, 3]-NRT-code), using
- appending kth column [i] based on linear OA(84, 585, F8, 2) (dual of [585, 581, 3]-code), using
- Hamming code H(4,8) [i]
- appending kth column [i] based on linear OA(84, 585, F8, 2) (dual of [585, 581, 3]-code), using
- OOA(819, 65281, S8, 2, 5), using
- OOA 2-folding [i] based on OA(819, 130562, S8, 5), using
- discarding parts of the base [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- discarding parts of the base [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- OOA 2-folding [i] based on OA(819, 130562, S8, 5), using
- linear OOA(84, 585, F8, 2, 2) (dual of [(585, 2), 1166, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on OOA(823, 65866, S8, 2, 5), using
(18, 18+5, 130834)-Net in Base 8
(18, 23, 130834)-net in base 8, using
- net defined by OOA [i] based on OOA(823, 130834, S8, 6, 5), using
- OOA stacking with additional row [i] based on OOA(823, 130835, S8, 2, 5), using
- discarding parts of the base [i] based on linear OOA(1617, 130835, F16, 2, 5) (dual of [(130835, 2), 261653, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1617, 130835, F16, 5) (dual of [130835, 130818, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(163, 273, F16, 2) (dual of [273, 270, 3]-code), using
- Hamming code H(3,16) [i]
- linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- linear OA(163, 273, F16, 2) (dual of [273, 270, 3]-code), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1617, 130835, F16, 5) (dual of [130835, 130818, 6]-code), using
- discarding parts of the base [i] based on linear OOA(1617, 130835, F16, 2, 5) (dual of [(130835, 2), 261653, 6]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(823, 130835, S8, 2, 5), using
(18, 18+5, large)-Net in Base 8 — Upper bound on s
There is no (18, 23, large)-net in base 8, because
- 3 times m-reduction [i] would yield (18, 20, large)-net in base 8, but