Best Known (81, 106, s)-Nets in Base 3
(81, 106, 400)-Net over F3 — Constructive and digital
Digital (81, 106, 400)-net over F3, using
- 32 times duplication [i] based on digital (79, 104, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 26, 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, 26, 100)-net over F81, using
(81, 106, 690)-Net over F3 — Digital
Digital (81, 106, 690)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3106, 690, F3, 25) (dual of [690, 584, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 763, F3, 25) (dual of [763, 657, 26]-code), using
- construction XX applied to C1 = C([722,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([722,18]) [i] based on
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,16}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(373, 728, F3, 19) (dual of [728, 655, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 728, F3, 25) (dual of [728, 631, 26]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−6,−5,…,18}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([722,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([722,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3106, 763, F3, 25) (dual of [763, 657, 26]-code), using
(81, 106, 39538)-Net in Base 3 — Upper bound on s
There is no (81, 106, 39539)-net in base 3, because
- 1 times m-reduction [i] would yield (81, 105, 39539)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 125 269844 439372 155106 558992 553728 217531 522474 434889 > 3105 [i]