Best Known (126−18, 126, s)-Nets in Base 3
(126−18, 126, 6564)-Net over F3 — Constructive and digital
Digital (108, 126, 6564)-net over F3, using
- net defined by OOA [i] based on linear OOA(3126, 6564, F3, 18, 18) (dual of [(6564, 18), 118026, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3126, 59076, F3, 18) (dual of [59076, 58950, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3126, 59077, F3, 18) (dual of [59077, 58951, 19]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3126, 59077, F3, 18) (dual of [59077, 58951, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3126, 59076, F3, 18) (dual of [59076, 58950, 19]-code), using
(126−18, 126, 28231)-Net over F3 — Digital
Digital (108, 126, 28231)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3126, 28231, F3, 2, 18) (dual of [(28231, 2), 56336, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3126, 29538, F3, 2, 18) (dual of [(29538, 2), 58950, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3126, 59076, F3, 18) (dual of [59076, 58950, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3126, 59077, F3, 18) (dual of [59077, 58951, 19]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3126, 59077, F3, 18) (dual of [59077, 58951, 19]-code), using
- OOA 2-folding [i] based on linear OA(3126, 59076, F3, 18) (dual of [59076, 58950, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(3126, 29538, F3, 2, 18) (dual of [(29538, 2), 58950, 19]-NRT-code), using
(126−18, 126, large)-Net in Base 3 — Upper bound on s
There is no (108, 126, large)-net in base 3, because
- 16 times m-reduction [i] would yield (108, 110, large)-net in base 3, but