Best Known (83−17, 83, s)-Nets in Base 3
(83−17, 83, 464)-Net over F3 — Constructive and digital
Digital (66, 83, 464)-net over F3, using
- t-expansion [i] based on digital (65, 83, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (65, 84, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 21, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 21, 116)-net over F81, using
- 1 times m-reduction [i] based on digital (65, 84, 464)-net over F3, using
(83−17, 83, 1290)-Net over F3 — Digital
Digital (66, 83, 1290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(383, 1290, F3, 17) (dual of [1290, 1207, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(383, 2207, F3, 17) (dual of [2207, 2124, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(357, 2187, F3, 13) (dual of [2187, 2130, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(383, 2207, F3, 17) (dual of [2207, 2124, 18]-code), using
(83−17, 83, 146261)-Net in Base 3 — Upper bound on s
There is no (66, 83, 146262)-net in base 3, because
- 1 times m-reduction [i] would yield (66, 82, 146262)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1330 300188 556450 126574 559187 422451 268817 > 382 [i]