Best Known (121, 121+27, s)-Nets in Base 3
(121, 121+27, 688)-Net over F3 — Constructive and digital
Digital (121, 148, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (121, 152, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 172)-net over F81, using
(121, 121+27, 3290)-Net over F3 — Digital
Digital (121, 148, 3290)-net over F3, using
- 31 times duplication [i] based on digital (120, 147, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 3290, F3, 2, 27) (dual of [(3290, 2), 6433, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3147, 6580, F3, 27) (dual of [6580, 6433, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3146, 6579, F3, 27) (dual of [6579, 6433, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3146, 6579, F3, 27) (dual of [6579, 6433, 28]-code), using
- OOA 2-folding [i] based on linear OA(3147, 6580, F3, 27) (dual of [6580, 6433, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 3290, F3, 2, 27) (dual of [(3290, 2), 6433, 28]-NRT-code), using
(121, 121+27, 703896)-Net in Base 3 — Upper bound on s
There is no (121, 148, 703897)-net in base 3, because
- 1 times m-reduction [i] would yield (121, 147, 703897)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13703 401140 771968 081628 900540 250642 778222 784270 180394 985167 029136 923651 > 3147 [i]