Best Known (54−9, 54, s)-Nets in Base 8
(54−9, 54, 524296)-Net over F8 — Constructive and digital
Digital (45, 54, 524296)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (41, 50, 524287)-net over F8, using
- net defined by OOA [i] based on linear OOA(850, 524287, F8, 9, 9) (dual of [(524287, 9), 4718533, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(850, 2097149, F8, 9) (dual of [2097149, 2097099, 10]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(850, 2097152, F8, 9) (dual of [2097152, 2097102, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(850, 2097149, F8, 9) (dual of [2097149, 2097099, 10]-code), using
- net defined by OOA [i] based on linear OOA(850, 524287, F8, 9, 9) (dual of [(524287, 9), 4718533, 10]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
(54−9, 54, 2097177)-Net over F8 — Digital
Digital (45, 54, 2097177)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(854, 2097177, F8, 9) (dual of [2097177, 2097123, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- 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(829, 2097152, F8, 5) (dual of [2097152, 2097123, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(84, 25, F8, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,8)), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
(54−9, 54, large)-Net in Base 8 — Upper bound on s
There is no (45, 54, large)-net in base 8, because
- 7 times m-reduction [i] would yield (45, 47, large)-net in base 8, but