Best Known (124−17, 124, s)-Nets in Base 8
(124−17, 124, 1048599)-Net over F8 — Constructive and digital
Digital (107, 124, 1048599)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- digital (96, 113, 1048575)-net over F8, using
- net defined by OOA [i] based on linear OOA(8113, 1048575, F8, 17, 17) (dual of [(1048575, 17), 17825662, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8113, 8388601, F8, 17) (dual of [8388601, 8388488, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8113, 8388601, F8, 17) (dual of [8388601, 8388488, 18]-code), using
- net defined by OOA [i] based on linear OOA(8113, 1048575, F8, 17, 17) (dual of [(1048575, 17), 17825662, 18]-NRT-code), using
- digital (3, 11, 24)-net over F8, using
(124−17, 124, large)-Net over F8 — Digital
Digital (107, 124, large)-net over F8, using
- t-expansion [i] based on digital (106, 124, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8124, large, F8, 18) (dual of [large, large−124, 19]-code), using
- 3 times code embedding in larger space [i] based on linear OA(8121, large, F8, 18) (dual of [large, large−121, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 3 times code embedding in larger space [i] based on linear OA(8121, large, F8, 18) (dual of [large, large−121, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8124, large, F8, 18) (dual of [large, large−124, 19]-code), using
(124−17, 124, large)-Net in Base 8 — Upper bound on s
There is no (107, 124, large)-net in base 8, because
- 15 times m-reduction [i] would yield (107, 109, large)-net in base 8, but