Best Known (99, 132, s)-Nets in Base 3
(99, 132, 328)-Net over F3 — Constructive and digital
Digital (99, 132, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 33, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
(99, 132, 618)-Net over F3 — Digital
Digital (99, 132, 618)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3132, 618, F3, 33) (dual of [618, 486, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3132, 748, F3, 33) (dual of [748, 616, 34]-code), using
- construction XX applied to C1 = C([725,28]), C2 = C([0,30]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([725,30]) [i] based on
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,28}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3118, 728, F3, 31) (dual of [728, 610, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3130, 728, F3, 34) (dual of [728, 598, 35]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,30}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- 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 (see above)
- construction XX applied to C1 = C([725,28]), C2 = C([0,30]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([725,30]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3132, 748, F3, 33) (dual of [748, 616, 34]-code), using
(99, 132, 27396)-Net in Base 3 — Upper bound on s
There is no (99, 132, 27397)-net in base 3, because
- 1 times m-reduction [i] would yield (99, 131, 27397)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 318 447410 422916 917331 887298 946143 680789 425348 284600 278073 353665 > 3131 [i]