Best Known (94−22, 94, s)-Nets in Base 3
(94−22, 94, 400)-Net over F3 — Constructive and digital
Digital (72, 94, 400)-net over F3, using
- 32 times duplication [i] based on digital (70, 92, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 23, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 23, 100)-net over F81, using
(94−22, 94, 669)-Net over F3 — Digital
Digital (72, 94, 669)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(394, 669, F3, 22) (dual of [669, 575, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(394, 763, F3, 22) (dual of [763, 669, 23]-code), using
- construction XX applied to C1 = C([722,13]), C2 = C([0,15]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([722,15]) [i] based on
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,13}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(361, 728, F3, 16) (dual of [728, 667, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(385, 728, F3, 22) (dual of [728, 643, 23]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,15}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(355, 728, F3, 14) (dual of [728, 673, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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
- construction XX applied to C1 = C([722,13]), C2 = C([0,15]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([722,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(394, 763, F3, 22) (dual of [763, 669, 23]-code), using
(94−22, 94, 29312)-Net in Base 3 — Upper bound on s
There is no (72, 94, 29313)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 707 209099 007138 964064 124734 860874 023726 529019 > 394 [i]