Best Known (142, 166, s)-Nets in Base 3
(142, 166, 4923)-Net over F3 — Constructive and digital
Digital (142, 166, 4923)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 4923, F3, 24, 24) (dual of [(4923, 24), 117986, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3166, 59076, F3, 24) (dual of [59076, 58910, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 59077, F3, 24) (dual of [59077, 58911, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 59077, F3, 24) (dual of [59077, 58911, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3166, 59076, F3, 24) (dual of [59076, 58910, 25]-code), using
(142, 166, 23077)-Net over F3 — Digital
Digital (142, 166, 23077)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3166, 23077, F3, 2, 24) (dual of [(23077, 2), 45988, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3166, 29538, F3, 2, 24) (dual of [(29538, 2), 58910, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3166, 59076, F3, 24) (dual of [59076, 58910, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 59077, F3, 24) (dual of [59077, 58911, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 59077, F3, 24) (dual of [59077, 58911, 25]-code), using
- OOA 2-folding [i] based on linear OA(3166, 59076, F3, 24) (dual of [59076, 58910, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(3166, 29538, F3, 2, 24) (dual of [(29538, 2), 58910, 25]-NRT-code), using
(142, 166, large)-Net in Base 3 — Upper bound on s
There is no (142, 166, large)-net in base 3, because
- 22 times m-reduction [i] would yield (142, 144, large)-net in base 3, but