Best Known (123, 123+30, s)-Nets in Base 3
(123, 123+30, 688)-Net over F3 — Constructive and digital
Digital (123, 153, 688)-net over F3, using
- 31 times duplication [i] based on digital (122, 152, 688)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (121, 152, 688)-net over F3, using
(123, 123+30, 2172)-Net over F3 — Digital
Digital (123, 153, 2172)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3153, 2172, F3, 30) (dual of [2172, 2019, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 2228, F3, 30) (dual of [2228, 2075, 31]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3149, 2224, F3, 30) (dual of [2224, 2075, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(38, 36, F3, 4) (dual of [36, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3149, 2224, F3, 30) (dual of [2224, 2075, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 2228, F3, 30) (dual of [2228, 2075, 31]-code), using
(123, 123+30, 236236)-Net in Base 3 — Upper bound on s
There is no (123, 153, 236237)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9 990079 228886 273066 689508 584362 082633 479326 482306 857228 031472 231976 484251 > 3153 [i]