Best Known (111−17, 111, s)-Nets in Base 8
(111−17, 111, 262169)-Net over F8 — Constructive and digital
Digital (94, 111, 262169)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
- net from sequence [i] based on digital (4, 24)-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 (4, 12, 25)-net over F8, using
(111−17, 111, 2097213)-Net over F8 — Digital
Digital (94, 111, 2097213)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8111, 2097213, F8, 17) (dual of [2097213, 2097102, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(8) [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(850, 2097152, F8, 9) (dual of [2097152, 2097102, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(812, 61, F8, 7) (dual of [61, 49, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(812, 73, F8, 7) (dual of [73, 61, 8]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(812, 73, F8, 7) (dual of [73, 61, 8]-code), using
- construction X applied to Ce(16) ⊂ Ce(8) [i] based on
(111−17, 111, large)-Net in Base 8 — Upper bound on s
There is no (94, 111, large)-net in base 8, because
- 15 times m-reduction [i] would yield (94, 96, large)-net in base 8, but