Best Known (79−17, 79, s)-Nets in Base 8
(79−17, 79, 4105)-Net over F8 — Constructive and digital
Digital (62, 79, 4105)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (54, 71, 4096)-net over F8, using
- net defined by OOA [i] based on linear OOA(871, 4096, F8, 17, 17) (dual of [(4096, 17), 69561, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [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]
- OOA 8-folding and stacking with additional row [i] based on linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using
- net defined by OOA [i] based on linear OOA(871, 4096, F8, 17, 17) (dual of [(4096, 17), 69561, 18]-NRT-code), using
- digital (0, 8, 9)-net over F8, using
(79−17, 79, 32802)-Net over F8 — Digital
Digital (62, 79, 32802)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(879, 32802, F8, 17) (dual of [32802, 32723, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(878, 32800, F8, 17) (dual of [32800, 32722, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [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(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(878, 32801, F8, 16) (dual of [32801, 32723, 17]-code), using Gilbert–Varšamov bound and bm = 878 > Vbs−1(k−1) = 198 023418 585156 007258 150284 859167 982640 685729 291439 038544 193125 469661 [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(878, 32800, F8, 17) (dual of [32800, 32722, 18]-code), using
- construction X with Varšamov bound [i] based on
(79−17, 79, large)-Net in Base 8 — Upper bound on s
There is no (62, 79, large)-net in base 8, because
- 15 times m-reduction [i] would yield (62, 64, large)-net in base 8, but