Best Known (239−37, 239, s)-Nets in Base 3
(239−37, 239, 1499)-Net over F3 — Constructive and digital
Digital (202, 239, 1499)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (9, 27, 19)-net over F3, using
- net from sequence [i] based on digital (9, 18)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using
- net from sequence [i] based on digital (9, 18)-sequence over F3, using
- digital (175, 212, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 53, 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, 53, 370)-net over F81, using
- digital (9, 27, 19)-net over F3, using
(239−37, 239, 12175)-Net over F3 — Digital
Digital (202, 239, 12175)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3239, 12175, F3, 37) (dual of [12175, 11936, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 19759, F3, 37) (dual of [19759, 19520, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3238, 19758, F3, 37) (dual of [19758, 19520, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(27) [i] based on
- linear OA(3217, 19683, F3, 37) (dual of [19683, 19466, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(321, 75, F3, 8) (dual of [75, 54, 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(36) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3238, 19758, F3, 37) (dual of [19758, 19520, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 19759, F3, 37) (dual of [19759, 19520, 38]-code), using
(239−37, 239, 7686034)-Net in Base 3 — Upper bound on s
There is no (202, 239, 7686035)-net in base 3, because
- 1 times m-reduction [i] would yield (202, 238, 7686035)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 358805 540563 351739 808834 699406 318773 388381 711928 578392 823135 828197 126081 359457 581376 363498 715677 538697 058669 711781 > 3238 [i]