Best Known (239−41, 239, s)-Nets in Base 3
(239−41, 239, 1480)-Net over F3 — Constructive and digital
Digital (198, 239, 1480)-net over F3, using
- t-expansion [i] based on digital (196, 239, 1480)-net over F3, using
- 1 times m-reduction [i] based on digital (196, 240, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 60, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 60, 370)-net over F81, using
- 1 times m-reduction [i] based on digital (196, 240, 1480)-net over F3, using
(239−41, 239, 6244)-Net over F3 — Digital
Digital (198, 239, 6244)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3239, 6244, F3, 41) (dual of [6244, 6005, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 6631, F3, 41) (dual of [6631, 6392, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3238, 6630, F3, 41) (dual of [6630, 6392, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(31) [i] based on
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(321, 69, F3, 8) (dual of [69, 48, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(40) ⊂ Ce(31) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3238, 6630, F3, 41) (dual of [6630, 6392, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 6631, F3, 41) (dual of [6631, 6392, 42]-code), using
(239−41, 239, 1977037)-Net in Base 3 — Upper bound on s
There is no (198, 239, 1977038)-net in base 3, because
- 1 times m-reduction [i] would yield (198, 238, 1977038)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 358807 327177 019534 233771 218130 071868 518823 290679 333706 837226 343695 382948 073673 576073 858016 217543 910482 828455 732649 > 3238 [i]