Best Known (107, 107+25, s)-Nets in Base 3
(107, 107+25, 688)-Net over F3 — Constructive and digital
Digital (107, 132, 688)-net over F3, using
- t-expansion [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
(107, 107+25, 2966)-Net over F3 — Digital
Digital (107, 132, 2966)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3132, 2966, F3, 2, 25) (dual of [(2966, 2), 5800, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3132, 3287, F3, 2, 25) (dual of [(3287, 2), 6442, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3132, 6574, F3, 25) (dual of [6574, 6442, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3132, 6574, F3, 25) (dual of [6574, 6442, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3132, 3287, F3, 2, 25) (dual of [(3287, 2), 6442, 26]-NRT-code), using
(107, 107+25, 427457)-Net in Base 3 — Upper bound on s
There is no (107, 132, 427458)-net in base 3, because
- 1 times m-reduction [i] would yield (107, 131, 427458)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 318 335139 362697 489897 419976 327307 430404 396080 915098 923167 731577 > 3131 [i]