Best Known (106, 106+23, s)-Nets in Base 3
(106, 106+23, 688)-Net over F3 — Constructive and digital
Digital (106, 129, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (106, 132, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 33, 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, 33, 172)-net over F81, using
(106, 106+23, 3493)-Net over F3 — Digital
Digital (106, 129, 3493)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3129, 3493, F3, 23) (dual of [3493, 3364, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 6594, F3, 23) (dual of [6594, 6465, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(22) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3129, 6594, F3, 23) (dual of [6594, 6465, 24]-code), using
(106, 106+23, 874853)-Net in Base 3 — Upper bound on s
There is no (106, 129, 874854)-net in base 3, because
- 1 times m-reduction [i] would yield (106, 128, 874854)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11 790248 815094 933660 789014 352762 425059 786595 745409 266187 899281 > 3128 [i]