Best Known (108, 108+17, s)-Nets in Base 3
(108, 108+17, 22145)-Net over F3 — Constructive and digital
Digital (108, 125, 22145)-net over F3, using
- net defined by OOA [i] based on linear OOA(3125, 22145, F3, 17, 17) (dual of [(22145, 17), 376340, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3125, 177161, F3, 17) (dual of [177161, 177036, 18]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3122, 177158, F3, 17) (dual of [177158, 177036, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3122, 177158, F3, 17) (dual of [177158, 177036, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3125, 177161, F3, 17) (dual of [177161, 177036, 18]-code), using
(108, 108+17, 59053)-Net over F3 — Digital
Digital (108, 125, 59053)-net over F3, using
- 32 times duplication [i] based on digital (106, 123, 59053)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3123, 59053, F3, 3, 17) (dual of [(59053, 3), 177036, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3123, 177159, F3, 17) (dual of [177159, 177036, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3122, 177158, F3, 17) (dual of [177158, 177036, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3111, 177147, F3, 16) (dual of [177147, 177036, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3122, 177158, F3, 17) (dual of [177158, 177036, 18]-code), using
- OOA 3-folding [i] based on linear OA(3123, 177159, F3, 17) (dual of [177159, 177036, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3123, 59053, F3, 3, 17) (dual of [(59053, 3), 177036, 18]-NRT-code), using
(108, 108+17, large)-Net in Base 3 — Upper bound on s
There is no (108, 125, large)-net in base 3, because
- 15 times m-reduction [i] would yield (108, 110, large)-net in base 3, but