Best Known (109, 109+18, s)-Nets in Base 3
(109, 109+18, 6565)-Net over F3 — Constructive and digital
Digital (109, 127, 6565)-net over F3, using
- net defined by OOA [i] based on linear OOA(3127, 6565, F3, 18, 18) (dual of [(6565, 18), 118043, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3127, 59085, F3, 18) (dual of [59085, 58958, 19]-code), using
- construction X applied to Ce(18) ⊂ 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(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(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OA 9-folding and stacking [i] based on linear OA(3127, 59085, F3, 18) (dual of [59085, 58958, 19]-code), using
(109, 109+18, 29542)-Net over F3 — Digital
Digital (109, 127, 29542)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3127, 29542, F3, 2, 18) (dual of [(29542, 2), 58957, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3127, 59084, F3, 18) (dual of [59084, 58957, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 59085, F3, 18) (dual of [59085, 58958, 19]-code), using
- construction X applied to Ce(18) ⊂ 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(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(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3127, 59085, F3, 18) (dual of [59085, 58958, 19]-code), using
- OOA 2-folding [i] based on linear OA(3127, 59084, F3, 18) (dual of [59084, 58957, 19]-code), using
(109, 109+18, large)-Net in Base 3 — Upper bound on s
There is no (109, 127, large)-net in base 3, because
- 16 times m-reduction [i] would yield (109, 111, large)-net in base 3, but