Best Known (249−35, 249, s)-Nets in Base 3
(249−35, 249, 3481)-Net over F3 — Constructive and digital
Digital (214, 249, 3481)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (196, 231, 3474)-net over F3, using
- net defined by OOA [i] based on linear OOA(3231, 3474, F3, 35, 35) (dual of [(3474, 35), 121359, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(3231, 59059, F3, 35) (dual of [59059, 58828, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3231, 59049, F3, 35) (dual of [59049, 58818, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- OOA 17-folding and stacking with additional row [i] based on linear OA(3231, 59059, F3, 35) (dual of [59059, 58828, 36]-code), using
- net defined by OOA [i] based on linear OOA(3231, 3474, F3, 35, 35) (dual of [(3474, 35), 121359, 36]-NRT-code), using
- digital (1, 18, 7)-net over F3, using
(249−35, 249, 29558)-Net over F3 — Digital
Digital (214, 249, 29558)-net over F3, using
- 31 times duplication [i] based on digital (213, 248, 29558)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3248, 29558, F3, 2, 35) (dual of [(29558, 2), 58868, 36]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3246, 29557, F3, 2, 35) (dual of [(29557, 2), 58868, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3246, 59114, F3, 35) (dual of [59114, 58868, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- linear OA(3231, 59049, F3, 35) (dual of [59049, 58818, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(315, 65, F3, 6) (dual of [65, 50, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3246, 59114, F3, 35) (dual of [59114, 58868, 36]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3246, 29557, F3, 2, 35) (dual of [(29557, 2), 58868, 36]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3248, 29558, F3, 2, 35) (dual of [(29558, 2), 58868, 36]-NRT-code), using
(249−35, 249, large)-Net in Base 3 — Upper bound on s
There is no (214, 249, large)-net in base 3, because
- 33 times m-reduction [i] would yield (214, 216, large)-net in base 3, but