Best Known (113−20, 113, s)-Nets in Base 9
(113−20, 113, 53154)-Net over F9 — Constructive and digital
Digital (93, 113, 53154)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (83, 103, 53144)-net over F9, using
- net defined by OOA [i] based on linear OOA(9103, 53144, F9, 20, 20) (dual of [(53144, 20), 1062777, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9103, 531440, F9, 20) (dual of [531440, 531337, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9103, 531440, F9, 20) (dual of [531440, 531337, 21]-code), using
- net defined by OOA [i] based on linear OOA(9103, 53144, F9, 20, 20) (dual of [(53144, 20), 1062777, 21]-NRT-code), using
- digital (0, 10, 10)-net over F9, using
(113−20, 113, 531487)-Net over F9 — Digital
Digital (93, 113, 531487)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9113, 531487, F9, 20) (dual of [531487, 531374, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(910, 46, F9, 6) (dual of [46, 36, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- a “Gra†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
(113−20, 113, large)-Net in Base 9 — Upper bound on s
There is no (93, 113, large)-net in base 9, because
- 18 times m-reduction [i] would yield (93, 95, large)-net in base 9, but