Best Known (90, 90+28, s)-Nets in Base 3
(90, 90+28, 400)-Net over F3 — Constructive and digital
Digital (90, 118, 400)-net over F3, using
- 32 times duplication [i] based on digital (88, 116, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 29, 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, 29, 100)-net over F81, using
(90, 90+28, 717)-Net over F3 — Digital
Digital (90, 118, 717)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3118, 717, F3, 28) (dual of [717, 599, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3118, 763, F3, 28) (dual of [763, 645, 29]-code), using
- construction XX applied to C1 = C([722,19]), C2 = C([0,21]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([722,21]) [i] based on
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,19}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(385, 728, F3, 22) (dual of [728, 643, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,21}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [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([722,19]), C2 = C([0,21]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([722,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3118, 763, F3, 28) (dual of [763, 645, 29]-code), using
(90, 90+28, 31746)-Net in Base 3 — Upper bound on s
There is no (90, 118, 31747)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 199 725611 087183 114783 809281 217593 194716 843901 884449 694797 > 3118 [i]