Best Known (104−15, 104, s)-Nets in Base 9
(104−15, 104, 683314)-Net over F9 — Constructive and digital
Digital (89, 104, 683314)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 32)-net over F9, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 5 and N(F) ≥ 32, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- digital (77, 92, 683282)-net over F9, using
- net defined by OOA [i] based on linear OOA(992, 683282, F9, 15, 15) (dual of [(683282, 15), 10249138, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(992, 4782975, F9, 15) (dual of [4782975, 4782883, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(992, 4782976, F9, 15) (dual of [4782976, 4782884, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(992, 4782976, F9, 15) (dual of [4782976, 4782884, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(992, 4782975, F9, 15) (dual of [4782975, 4782883, 16]-code), using
- net defined by OOA [i] based on linear OOA(992, 683282, F9, 15, 15) (dual of [(683282, 15), 10249138, 16]-NRT-code), using
- digital (5, 12, 32)-net over F9, using
(104−15, 104, 683320)-Net in Base 9 — Constructive
(89, 104, 683320)-net in base 9, using
- (u, u+v)-construction [i] based on
- (5, 12, 38)-net in base 9, using
- base change [i] based on digital (1, 8, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- base change [i] based on digital (1, 8, 38)-net over F27, using
- digital (77, 92, 683282)-net over F9, using
- net defined by OOA [i] based on linear OOA(992, 683282, F9, 15, 15) (dual of [(683282, 15), 10249138, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(992, 4782975, F9, 15) (dual of [4782975, 4782883, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(992, 4782976, F9, 15) (dual of [4782976, 4782884, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(992, 4782976, F9, 15) (dual of [4782976, 4782884, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(992, 4782975, F9, 15) (dual of [4782975, 4782883, 16]-code), using
- net defined by OOA [i] based on linear OOA(992, 683282, F9, 15, 15) (dual of [(683282, 15), 10249138, 16]-NRT-code), using
- (5, 12, 38)-net in base 9, using
(104−15, 104, large)-Net over F9 — Digital
Digital (89, 104, large)-net over F9, using
(104−15, 104, large)-Net in Base 9 — Upper bound on s
There is no (89, 104, large)-net in base 9, because
- 13 times m-reduction [i] would yield (89, 91, large)-net in base 9, but