Best Known (118−21, 118, s)-Nets in Base 3
(118−21, 118, 688)-Net over F3 — Constructive and digital
Digital (97, 118, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (97, 120, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 30, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 30, 172)-net over F81, using
(118−21, 118, 3420)-Net over F3 — Digital
Digital (97, 118, 3420)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3118, 3420, F3, 21) (dual of [3420, 3302, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3118, 6585, F3, 21) (dual of [6585, 6467, 22]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 4, F3, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3118, 6585, F3, 21) (dual of [6585, 6467, 22]-code), using
(118−21, 118, 865486)-Net in Base 3 — Upper bound on s
There is no (97, 118, 865487)-net in base 3, because
- 1 times m-reduction [i] would yield (97, 117, 865487)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 66 556439 845848 879443 908840 650243 037654 165516 804646 858637 > 3117 [i]