Best Known (217−32, 217, s)-Nets in Base 3
(217−32, 217, 3692)-Net over F3 — Constructive and digital
Digital (185, 217, 3692)-net over F3, using
- 32 times duplication [i] based on digital (183, 215, 3692)-net over F3, using
- net defined by OOA [i] based on linear OOA(3215, 3692, F3, 32, 32) (dual of [(3692, 32), 117929, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3215, 59072, F3, 32) (dual of [59072, 58857, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 59073, F3, 32) (dual of [59073, 58858, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3215, 59073, F3, 32) (dual of [59073, 58858, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3215, 59072, F3, 32) (dual of [59072, 58857, 33]-code), using
- net defined by OOA [i] based on linear OOA(3215, 3692, F3, 32, 32) (dual of [(3692, 32), 117929, 33]-NRT-code), using
(217−32, 217, 20080)-Net over F3 — Digital
Digital (185, 217, 20080)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 20080, F3, 2, 32) (dual of [(20080, 2), 39943, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3217, 29542, F3, 2, 32) (dual of [(29542, 2), 58867, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3217, 59084, F3, 32) (dual of [59084, 58867, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 59085, F3, 32) (dual of [59085, 58868, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(31) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 59085, F3, 32) (dual of [59085, 58868, 33]-code), using
- OOA 2-folding [i] based on linear OA(3217, 59084, F3, 32) (dual of [59084, 58867, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(3217, 29542, F3, 2, 32) (dual of [(29542, 2), 58867, 33]-NRT-code), using
(217−32, 217, large)-Net in Base 3 — Upper bound on s
There is no (185, 217, large)-net in base 3, because
- 30 times m-reduction [i] would yield (185, 187, large)-net in base 3, but