Best Known (138, 138+38, s)-Nets in Base 3
(138, 138+38, 640)-Net over F3 — Constructive and digital
Digital (138, 176, 640)-net over F3, using
- t-expansion [i] based on digital (137, 176, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 44, 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, 44, 160)-net over F81, using
(138, 138+38, 1457)-Net over F3 — Digital
Digital (138, 176, 1457)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3176, 1457, F3, 38) (dual of [1457, 1281, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using
- an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- discarding factors / shortening the dual code based on linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using
(138, 138+38, 104173)-Net in Base 3 — Upper bound on s
There is no (138, 176, 104174)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 940519 710848 858914 668169 541279 224723 516197 939941 524551 337636 466664 873082 550010 664353 > 3176 [i]