Best Known (38, 45, s)-Nets in Base 8
(38, 45, 699056)-Net over F8 — Constructive and digital
Digital (38, 45, 699056)-net over F8, using
- 81 times duplication [i] based on digital (37, 44, 699056)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 699056, F8, 7, 7) (dual of [(699056, 7), 4893348, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(844, 2097169, F8, 7) (dual of [2097169, 2097125, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(843, 2097153, F8, 7) (dual of [2097153, 2097110, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(829, 2097153, F8, 5) (dual of [2097153, 2097124, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(844, 2097169, F8, 7) (dual of [2097169, 2097125, 8]-code), using
- net defined by OOA [i] based on linear OOA(844, 699056, F8, 7, 7) (dual of [(699056, 7), 4893348, 8]-NRT-code), using
(38, 45, 2536877)-Net over F8 — Digital
Digital (38, 45, 2536877)-net over F8, using
(38, 45, large)-Net in Base 8 — Upper bound on s
There is no (38, 45, large)-net in base 8, because
- 5 times m-reduction [i] would yield (38, 40, large)-net in base 8, but