Best Known (61−9, 61, s)-Nets in Base 3
(61−9, 61, 14764)-Net over F3 — Constructive and digital
Digital (52, 61, 14764)-net over F3, using
- net defined by OOA [i] based on linear OOA(361, 14764, F3, 9, 9) (dual of [(14764, 9), 132815, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(361, 59057, F3, 9) (dual of [59057, 58996, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 59060, F3, 9) (dual of [59060, 58999, 10]-code), using
- 1 times truncation [i] based on linear OA(362, 59061, F3, 10) (dual of [59061, 58999, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(362, 59061, F3, 10) (dual of [59061, 58999, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 59060, F3, 9) (dual of [59060, 58999, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(361, 59057, F3, 9) (dual of [59057, 58996, 10]-code), using
(61−9, 61, 29530)-Net over F3 — Digital
Digital (52, 61, 29530)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(361, 29530, F3, 2, 9) (dual of [(29530, 2), 58999, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(361, 59060, F3, 9) (dual of [59060, 58999, 10]-code), using
- 1 times truncation [i] based on linear OA(362, 59061, F3, 10) (dual of [59061, 58999, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(362, 59061, F3, 10) (dual of [59061, 58999, 11]-code), using
- OOA 2-folding [i] based on linear OA(361, 59060, F3, 9) (dual of [59060, 58999, 10]-code), using
(61−9, 61, large)-Net in Base 3 — Upper bound on s
There is no (52, 61, large)-net in base 3, because
- 7 times m-reduction [i] would yield (52, 54, large)-net in base 3, but