Best Known (238−44, 238, s)-Nets in Base 3
(238−44, 238, 896)-Net over F3 — Constructive and digital
Digital (194, 238, 896)-net over F3, using
- t-expansion [i] based on digital (193, 238, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (193, 240, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 60, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 60, 224)-net over F81, using
- 2 times m-reduction [i] based on digital (193, 240, 896)-net over F3, using
(238−44, 238, 4025)-Net over F3 — Digital
Digital (194, 238, 4025)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3238, 4025, F3, 44) (dual of [4025, 3787, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 6583, F3, 44) (dual of [6583, 6345, 45]-code), using
- construction XX applied to Ce(43) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3233, 6561, F3, 44) (dual of [6561, 6328, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- 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(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(34, 21, F3, 2) (dual of [21, 17, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), 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(43) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 6583, F3, 44) (dual of [6583, 6345, 45]-code), using
(238−44, 238, 656739)-Net in Base 3 — Upper bound on s
There is no (194, 238, 656740)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 358808 234940 695045 889823 487391 384112 700790 457717 184589 282236 955556 092471 040069 509049 542697 532801 276896 793698 241049 > 3238 [i]