Best Known (108−17, 108, s)-Nets in Base 8
(108−17, 108, 262158)-Net over F8 — Constructive and digital
Digital (91, 108, 262158)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 14)-net over F8, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 1 and N(F) ≥ 14, using
- net from sequence [i] based on digital (1, 13)-sequence over F8, using
- digital (82, 99, 262144)-net over F8, using
- net defined by OOA [i] based on linear OOA(899, 262144, F8, 17, 17) (dual of [(262144, 17), 4456349, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, 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(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using
- net defined by OOA [i] based on linear OOA(899, 262144, F8, 17, 17) (dual of [(262144, 17), 4456349, 18]-NRT-code), using
- digital (1, 9, 14)-net over F8, using
(108−17, 108, 2097198)-Net over F8 — Digital
Digital (91, 108, 2097198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8108, 2097198, F8, 17) (dual of [2097198, 2097090, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8106, 2097194, F8, 17) (dual of [2097194, 2097088, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(864, 2097152, F8, 11) (dual of [2097152, 2097088, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 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(8106, 2097196, F8, 16) (dual of [2097196, 2097090, 17]-code), using Gilbert–Varšamov bound and bm = 8106 > Vbs−1(k−1) = 242399 596970 086351 243054 889726 218444 646027 501575 904216 113648 310799 616879 859753 883161 980522 962720 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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(8106, 2097194, F8, 17) (dual of [2097194, 2097088, 18]-code), using
- construction X with Varšamov bound [i] based on
(108−17, 108, large)-Net in Base 8 — Upper bound on s
There is no (91, 108, large)-net in base 8, because
- 15 times m-reduction [i] would yield (91, 93, large)-net in base 8, but