Best Known (86−12, 86, s)-Nets in Base 3
(86−12, 86, 9846)-Net over F3 — Constructive and digital
Digital (74, 86, 9846)-net over F3, using
- net defined by OOA [i] based on linear OOA(386, 9846, F3, 12, 12) (dual of [(9846, 12), 118066, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(386, 59076, F3, 12) (dual of [59076, 58990, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(386, 59077, F3, 12) (dual of [59077, 58991, 13]-code), using
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(31, 24, F3, 1) (dual of [24, 23, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(386, 59077, F3, 12) (dual of [59077, 58991, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(386, 59076, F3, 12) (dual of [59076, 58990, 13]-code), using
(86−12, 86, 29538)-Net over F3 — Digital
Digital (74, 86, 29538)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(386, 29538, F3, 2, 12) (dual of [(29538, 2), 58990, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(386, 59076, F3, 12) (dual of [59076, 58990, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(386, 59077, F3, 12) (dual of [59077, 58991, 13]-code), using
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(31, 24, F3, 1) (dual of [24, 23, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(386, 59077, F3, 12) (dual of [59077, 58991, 13]-code), using
- OOA 2-folding [i] based on linear OA(386, 59076, F3, 12) (dual of [59076, 58990, 13]-code), using
(86−12, 86, large)-Net in Base 3 — Upper bound on s
There is no (74, 86, large)-net in base 3, because
- 10 times m-reduction [i] would yield (74, 76, large)-net in base 3, but