Best Known (115−21, 115, s)-Nets in Base 3
(115−21, 115, 688)-Net over F3 — Constructive and digital
Digital (94, 115, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (94, 116, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 29, 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, 29, 172)-net over F81, using
(115−21, 115, 3290)-Net over F3 — Digital
Digital (94, 115, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3115, 3290, F3, 2, 21) (dual of [(3290, 2), 6465, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3115, 6580, F3, 21) (dual of [6580, 6465, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3114, 6579, F3, 21) (dual of [6579, 6465, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3114, 6579, F3, 21) (dual of [6579, 6465, 22]-code), using
- OOA 2-folding [i] based on linear OA(3115, 6580, F3, 21) (dual of [6580, 6465, 22]-code), using
(115−21, 115, 622475)-Net in Base 3 — Upper bound on s
There is no (94, 115, 622476)-net in base 3, because
- 1 times m-reduction [i] would yield (94, 114, 622476)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 465070 354922 885119 097773 484684 045841 923535 276058 316649 > 3114 [i]