Best Known (123−30, 123, s)-Nets in Base 3
(123−30, 123, 328)-Net over F3 — Constructive and digital
Digital (93, 123, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (93, 124, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 82)-net over F81, using
(123−30, 123, 653)-Net over F3 — Digital
Digital (93, 123, 653)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3123, 653, F3, 30) (dual of [653, 530, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3123, 749, F3, 30) (dual of [749, 626, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3103, 729, F3, 26) (dual of [729, 626, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3123, 749, F3, 30) (dual of [749, 626, 31]-code), using
(123−30, 123, 26235)-Net in Base 3 — Upper bound on s
There is no (93, 123, 26236)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 48530 323964 057092 752142 806099 458306 103993 825119 420827 685841 > 3123 [i]