Best Known (250−36, 250, s)-Nets in Base 3
(250−36, 250, 3283)-Net over F3 — Constructive and digital
Digital (214, 250, 3283)-net over F3, using
- net defined by OOA [i] based on linear OOA(3250, 3283, F3, 36, 36) (dual of [(3283, 36), 117938, 37]-NRT-code), using
- OA 18-folding and stacking [i] based on linear OA(3250, 59094, F3, 36) (dual of [59094, 58844, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 59099, F3, 36) (dual of [59099, 58849, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [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(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(39, 49, F3, 4) (dual of [49, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 59099, F3, 36) (dual of [59099, 58849, 37]-code), using
- OA 18-folding and stacking [i] based on linear OA(3250, 59094, F3, 36) (dual of [59094, 58844, 37]-code), using
(250−36, 250, 25317)-Net over F3 — Digital
Digital (214, 250, 25317)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3250, 25317, F3, 2, 36) (dual of [(25317, 2), 50384, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3250, 29549, F3, 2, 36) (dual of [(29549, 2), 58848, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3250, 59098, F3, 36) (dual of [59098, 58848, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 59099, F3, 36) (dual of [59099, 58849, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [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(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(39, 49, F3, 4) (dual of [49, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 59099, F3, 36) (dual of [59099, 58849, 37]-code), using
- OOA 2-folding [i] based on linear OA(3250, 59098, F3, 36) (dual of [59098, 58848, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(3250, 29549, F3, 2, 36) (dual of [(29549, 2), 58848, 37]-NRT-code), using
(250−36, 250, large)-Net in Base 3 — Upper bound on s
There is no (214, 250, large)-net in base 3, because
- 34 times m-reduction [i] would yield (214, 216, large)-net in base 3, but