Best Known (104−10, 104, s)-Nets in Base 9
(104−10, 104, 5746932)-Net over F9 — Constructive and digital
Digital (94, 104, 5746932)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (25, 30, 2391492)-net over F9, using
- net defined by OOA [i] based on linear OOA(930, 2391492, F9, 5, 5) (dual of [(2391492, 5), 11957430, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(930, 4782985, F9, 5) (dual of [4782985, 4782955, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(929, 4782970, F9, 5) (dual of [4782970, 4782941, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(915, 4782970, F9, 3) (dual of [4782970, 4782955, 4]-code or 4782970-cap in PG(14,9)), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(930, 4782985, F9, 5) (dual of [4782985, 4782955, 6]-code), using
- net defined by OOA [i] based on linear OOA(930, 2391492, F9, 5, 5) (dual of [(2391492, 5), 11957430, 6]-NRT-code), using
- digital (64, 74, 3355440)-net over F9, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F81, using
- digital (25, 30, 2391492)-net over F9, using
(104−10, 104, large)-Net over F9 — Digital
Digital (94, 104, large)-net over F9, using
- t-expansion [i] based on digital (89, 104, large)-net over F9, using
(104−10, 104, large)-Net in Base 9 — Upper bound on s
There is no (94, 104, large)-net in base 9, because
- 8 times m-reduction [i] would yield (94, 96, large)-net in base 9, but