Best Known (62, 62+9, s)-Nets in Base 3
(62, 62+9, 44294)-Net over F3 — Constructive and digital
Digital (62, 71, 44294)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 8)-net over F3, using
- digital (57, 66, 44286)-net over F3, using
- net defined by OOA [i] based on linear OOA(366, 44286, F3, 9, 9) (dual of [(44286, 9), 398508, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(366, 177145, F3, 9) (dual of [177145, 177079, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(366, 177146, F3, 9) (dual of [177146, 177080, 10]-code), using
- 1 times truncation [i] based on linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(366, 177146, F3, 9) (dual of [177146, 177080, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(366, 177145, F3, 9) (dual of [177145, 177079, 10]-code), using
- net defined by OOA [i] based on linear OOA(366, 44286, F3, 9, 9) (dual of [(44286, 9), 398508, 10]-NRT-code), using
(62, 62+9, 99787)-Net over F3 — Digital
Digital (62, 71, 99787)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(371, 99787, F3, 9) (dual of [99787, 99716, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(371, 177153, F3, 9) (dual of [177153, 177082, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([1,4]) [i] based on
- linear OA(367, 177148, F3, 9) (dual of [177148, 177081, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(366, 177148, F3, 4) (dual of [177148, 177082, 5]-code), using the narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(34, 5, F3, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,3)), using
- dual of repetition code with length 5 [i]
- construction X applied to C([0,4]) ⊂ C([1,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(371, 177153, F3, 9) (dual of [177153, 177082, 10]-code), using
(62, 62+9, large)-Net in Base 3 — Upper bound on s
There is no (62, 71, large)-net in base 3, because
- 7 times m-reduction [i] would yield (62, 64, large)-net in base 3, but