Best Known (70−11, 70, s)-Nets in Base 3
(70−11, 70, 3945)-Net over F3 — Constructive and digital
Digital (59, 70, 3945)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (53, 64, 3938)-net over F3, using
- net defined by OOA [i] based on linear OOA(364, 3938, F3, 11, 11) (dual of [(3938, 11), 43254, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(364, 19691, F3, 11) (dual of [19691, 19627, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(364, 19692, F3, 11) (dual of [19692, 19628, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(364, 19683, F3, 11) (dual of [19683, 19619, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(364, 19692, F3, 11) (dual of [19692, 19628, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(364, 19691, F3, 11) (dual of [19691, 19627, 12]-code), using
- net defined by OOA [i] based on linear OOA(364, 3938, F3, 11, 11) (dual of [(3938, 11), 43254, 12]-NRT-code), using
- digital (1, 6, 7)-net over F3, using
(70−11, 70, 9858)-Net over F3 — Digital
Digital (59, 70, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(370, 9858, F3, 2, 11) (dual of [(9858, 2), 19646, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(370, 19716, F3, 11) (dual of [19716, 19646, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(364, 19683, F3, 11) (dual of [19683, 19619, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(337, 19683, F3, 7) (dual of [19683, 19646, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-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 Ce(10) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(370, 19716, F3, 11) (dual of [19716, 19646, 12]-code), using
(70−11, 70, 5001257)-Net in Base 3 — Upper bound on s
There is no (59, 70, 5001258)-net in base 3, because
- 1 times m-reduction [i] would yield (59, 69, 5001258)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 834 385948 101502 974248 805316 168365 > 369 [i]