Best Known (118, 118+31, s)-Nets in Base 3
(118, 118+31, 640)-Net over F3 — Constructive and digital
Digital (118, 149, 640)-net over F3, using
- 31 times duplication [i] based on digital (117, 148, 640)-net over F3, using
- t-expansion [i] based on digital (116, 148, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F81, using
- t-expansion [i] based on digital (116, 148, 640)-net over F3, using
(118, 118+31, 1562)-Net over F3 — Digital
Digital (118, 149, 1562)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3149, 1562, F3, 31) (dual of [1562, 1413, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3149, 2216, F3, 31) (dual of [2216, 2067, 32]-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, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3149, 2216, F3, 31) (dual of [2216, 2067, 32]-code), using
(118, 118+31, 163792)-Net in Base 3 — Upper bound on s
There is no (118, 149, 163793)-net in base 3, because
- 1 times m-reduction [i] would yield (118, 148, 163793)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 41110 904894 579453 042144 271914 133926 267338 654608 913035 648111 522988 111275 > 3148 [i]