Best Known (118, 118+30, s)-Nets in Base 3
(118, 118+30, 688)-Net over F3 — Constructive and digital
Digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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
(118, 118+30, 1781)-Net over F3 — Digital
Digital (118, 148, 1781)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3148, 1781, F3, 30) (dual of [1781, 1633, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3148, 2216, F3, 30) (dual of [2216, 2068, 31]-code), using
- construction XX applied to Ce(30) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3148, 2216, F3, 30) (dual of [2216, 2068, 31]-code), using
(118, 118+30, 163792)-Net in Base 3 — Upper bound on s
There is no (118, 148, 163793)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 41110 904894 579453 042144 271914 133926 267338 654608 913035 648111 522988 111275 > 3148 [i]