Best Known (239−46, 239, s)-Nets in Base 3
(239−46, 239, 896)-Net over F3 — Constructive and digital
Digital (193, 239, 896)-net over F3, using
- 1 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
(239−46, 239, 3245)-Net over F3 — Digital
Digital (193, 239, 3245)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3239, 3245, F3, 46) (dual of [3245, 3006, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, 3292, F3, 46) (dual of [3292, 3053, 47]-code), using
- construction XX applied to Ce(45) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3237, 3281, F3, 46) (dual of [3281, 3044, 47]-code), using an extension Ce(45) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3229, 3281, F3, 44) (dual of [3281, 3052, 45]-code), using an extension Ce(43) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3221, 3281, F3, 43) (dual of [3281, 3060, 44]-code), using an extension Ce(42) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, 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(45) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(3239, 3292, F3, 46) (dual of [3292, 3053, 47]-code), using
(239−46, 239, 427873)-Net in Base 3 — Upper bound on s
There is no (193, 239, 427874)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 076431 370843 441459 648173 071706 984429 841700 242878 927782 508267 124111 788124 194187 787441 087728 597962 788060 599503 653705 > 3239 [i]