Best Known (250−46, 250, s)-Nets in Base 3
(250−46, 250, 1480)-Net over F3 — Constructive and digital
Digital (204, 250, 1480)-net over F3, using
- 32 times duplication [i] based on digital (202, 248, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 62, 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, 62, 370)-net over F81, using
(250−46, 250, 4283)-Net over F3 — Digital
Digital (204, 250, 4283)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3250, 4283, F3, 46) (dual of [4283, 4033, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 6595, F3, 46) (dual of [6595, 6345, 47]-code), using
- construction XX applied to Ce(45) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3241, 6561, F3, 46) (dual of [6561, 6320, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [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(38, 33, F3, 4) (dual of [33, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-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(45) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 6595, F3, 46) (dual of [6595, 6345, 47]-code), using
(250−46, 250, 723624)-Net in Base 3 — Upper bound on s
There is no (204, 250, 723625)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 190685 639491 607140 438046 873482 599675 982912 928515 594762 688695 110693 089787 397821 660759 653484 984249 137974 097856 053429 379051 > 3250 [i]