Best Known (206−35, 206, s)-Nets in Base 3
(206−35, 206, 1480)-Net over F3 — Constructive and digital
Digital (171, 206, 1480)-net over F3, using
- 32 times duplication [i] based on digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
(206−35, 206, 6025)-Net over F3 — Digital
Digital (171, 206, 6025)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3206, 6025, F3, 35) (dual of [6025, 5819, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3206, 6630, F3, 35) (dual of [6630, 6424, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(321, 69, F3, 8) (dual of [69, 48, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3206, 6630, F3, 35) (dual of [6630, 6424, 36]-code), using
(206−35, 206, 2034382)-Net in Base 3 — Upper bound on s
There is no (171, 206, 2034383)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 205, 2034383)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 64 544241 014714 632702 385321 860177 782026 147745 357245 919580 976026 226095 027475 687305 735441 887467 459967 > 3205 [i]