Best Known (250−37, 250, s)-Nets in Base 3
(250−37, 250, 3282)-Net over F3 — Constructive and digital
Digital (213, 250, 3282)-net over F3, using
- 32 times duplication [i] based on digital (211, 248, 3282)-net over F3, using
- net defined by OOA [i] based on linear OOA(3248, 3282, F3, 37, 37) (dual of [(3282, 37), 121186, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(3248, 59077, F3, 37) (dual of [59077, 58829, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3247, 59076, F3, 37) (dual of [59076, 58829, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3241, 59050, F3, 37) (dual of [59050, 58809, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3221, 59050, F3, 33) (dual of [59050, 58829, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3247, 59076, F3, 37) (dual of [59076, 58829, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(3248, 59077, F3, 37) (dual of [59077, 58829, 38]-code), using
- net defined by OOA [i] based on linear OOA(3248, 3282, F3, 37, 37) (dual of [(3282, 37), 121186, 38]-NRT-code), using
(250−37, 250, 20414)-Net over F3 — Digital
Digital (213, 250, 20414)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3250, 20414, F3, 2, 37) (dual of [(20414, 2), 40578, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3250, 29544, F3, 2, 37) (dual of [(29544, 2), 58838, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3250, 59088, F3, 37) (dual of [59088, 58838, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3249, 59087, F3, 37) (dual of [59087, 58838, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(3241, 59049, F3, 37) (dual of [59049, 58808, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3211, 59049, F3, 32) (dual of [59049, 58838, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 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
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3249, 59087, F3, 37) (dual of [59087, 58838, 38]-code), using
- OOA 2-folding [i] based on linear OA(3250, 59088, F3, 37) (dual of [59088, 58838, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(3250, 29544, F3, 2, 37) (dual of [(29544, 2), 58838, 38]-NRT-code), using
(250−37, 250, large)-Net in Base 3 — Upper bound on s
There is no (213, 250, large)-net in base 3, because
- 35 times m-reduction [i] would yield (213, 215, large)-net in base 3, but