Best Known (194, 194+46, s)-Nets in Base 3
(194, 194+46, 896)-Net over F3 — Constructive and digital
Digital (194, 240, 896)-net over F3, using
- t-expansion [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
(194, 194+46, 3296)-Net over F3 — Digital
Digital (194, 240, 3296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3240, 3296, F3, 46) (dual of [3296, 3056, 47]-code), using
- 3 step Varšamov–Edel lengthening with (ri) = (1, 0, 0) [i] 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
- 3 step Varšamov–Edel lengthening with (ri) = (1, 0, 0) [i] based on linear OA(3239, 3292, F3, 46) (dual of [3292, 3053, 47]-code), using
(194, 194+46, 448808)-Net in Base 3 — Upper bound on s
There is no (194, 240, 448809)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 229327 047710 909790 891838 002708 702714 372598 255918 328593 699469 006714 618576 477964 025661 009611 006245 461925 318811 900395 > 3240 [i]