Best Known (59, 59+20, s)-Nets in Base 3
(59, 59+20, 228)-Net over F3 — Constructive and digital
Digital (59, 79, 228)-net over F3, using
- 31 times duplication [i] based on digital (58, 78, 228)-net over F3, using
- trace code for nets [i] based on digital (6, 26, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- trace code for nets [i] based on digital (6, 26, 76)-net over F27, using
(59, 59+20, 426)-Net over F3 — Digital
Digital (59, 79, 426)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(379, 426, F3, 20) (dual of [426, 347, 21]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using
(59, 59+20, 13301)-Net in Base 3 — Upper bound on s
There is no (59, 79, 13302)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 49 295350 715673 340663 811269 556427 214109 > 379 [i]