Best Known (104, 104+34, s)-Nets in Base 3
(104, 104+34, 328)-Net over F3 — Constructive and digital
Digital (104, 138, 328)-net over F3, using
- 32 times duplication [i] based on digital (102, 136, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 34, 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
- trace code for nets [i] based on digital (0, 34, 82)-net over F81, using
(104, 104+34, 678)-Net over F3 — Digital
Digital (104, 138, 678)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3138, 678, F3, 34) (dual of [678, 540, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 757, F3, 34) (dual of [757, 619, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3109, 729, F3, 28) (dual of [729, 620, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3138, 757, F3, 34) (dual of [757, 619, 35]-code), using
(104, 104+34, 26776)-Net in Base 3 — Upper bound on s
There is no (104, 138, 26777)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 696531 667044 975433 351873 876045 724129 343603 680793 981612 830794 047987 > 3138 [i]