Best Known (30−7, 30, s)-Nets in Base 9
(30−7, 30, 4457)-Net over F9 — Constructive and digital
Digital (23, 30, 4457)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 82)-net over F9, using
- net defined by OOA [i] based on linear OOA(94, 82, F9, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
- digital (19, 26, 4375)-net over F9, using
- net defined by OOA [i] based on linear OOA(926, 4375, F9, 7, 7) (dual of [(4375, 7), 30599, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(926, 13126, F9, 7) (dual of [13126, 13100, 8]-code), using
- trace code [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(926, 13126, F9, 7) (dual of [13126, 13100, 8]-code), using
- net defined by OOA [i] based on linear OOA(926, 4375, F9, 7, 7) (dual of [(4375, 7), 30599, 8]-NRT-code), using
- digital (1, 4, 82)-net over F9, using
(30−7, 30, 6563)-Net in Base 9 — Constructive
(23, 30, 6563)-net in base 9, using
- base change [i] based on digital (13, 20, 6563)-net over F27, using
- net defined by OOA [i] based on linear OOA(2720, 6563, F27, 7, 7) (dual of [(6563, 7), 45921, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2720, 19690, F27, 7) (dual of [19690, 19670, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2720, 19691, F27, 7) (dual of [19691, 19671, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2719, 19684, F27, 7) (dual of [19684, 19665, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2713, 19684, F27, 5) (dual of [19684, 19671, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2720, 19691, F27, 7) (dual of [19691, 19671, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2720, 19690, F27, 7) (dual of [19690, 19670, 8]-code), using
- net defined by OOA [i] based on linear OOA(2720, 6563, F27, 7, 7) (dual of [(6563, 7), 45921, 8]-NRT-code), using
(30−7, 30, 22100)-Net over F9 — Digital
Digital (23, 30, 22100)-net over F9, using
(30−7, 30, large)-Net in Base 9 — Upper bound on s
There is no (23, 30, large)-net in base 9, because
- 5 times m-reduction [i] would yield (23, 25, large)-net in base 9, but