Best Known (237−24, 237, s)-Nets in Base 3
(237−24, 237, 398587)-Net over F3 — Constructive and digital
Digital (213, 237, 398587)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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 (200, 224, 398580)-net over F3, using
- net defined by OOA [i] based on linear OOA(3224, 398580, F3, 24, 24) (dual of [(398580, 24), 9565696, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3224, 4782960, F3, 24) (dual of [4782960, 4782736, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 4782969, F3, 24) (dual of [4782969, 4782745, 25]-code), using
- 1 times truncation [i] based on linear OA(3225, 4782970, F3, 25) (dual of [4782970, 4782745, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3225, 4782970, F3, 25) (dual of [4782970, 4782745, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 4782969, F3, 24) (dual of [4782969, 4782745, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3224, 4782960, F3, 24) (dual of [4782960, 4782736, 25]-code), using
- net defined by OOA [i] based on linear OOA(3224, 398580, F3, 24, 24) (dual of [(398580, 24), 9565696, 25]-NRT-code), using
- digital (1, 13, 7)-net over F3, using
(237−24, 237, 1587047)-Net over F3 — Digital
Digital (213, 237, 1587047)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3237, 1587047, F3, 3, 24) (dual of [(1587047, 3), 4760904, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3237, 1594346, F3, 3, 24) (dual of [(1594346, 3), 4782801, 25]-NRT-code), using
- strength reduction [i] based on linear OOA(3237, 1594346, F3, 3, 25) (dual of [(1594346, 3), 4782801, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3237, 4783038, F3, 25) (dual of [4783038, 4782801, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3236, 4783037, F3, 25) (dual of [4783037, 4782801, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3225, 4782970, F3, 25) (dual of [4782970, 4782745, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(311, 67, F3, 5) (dual of [67, 56, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3236, 4783037, F3, 25) (dual of [4783037, 4782801, 26]-code), using
- OOA 3-folding [i] based on linear OA(3237, 4783038, F3, 25) (dual of [4783038, 4782801, 26]-code), using
- strength reduction [i] based on linear OOA(3237, 1594346, F3, 3, 25) (dual of [(1594346, 3), 4782801, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3237, 1594346, F3, 3, 24) (dual of [(1594346, 3), 4782801, 25]-NRT-code), using
(237−24, 237, large)-Net in Base 3 — Upper bound on s
There is no (213, 237, large)-net in base 3, because
- 22 times m-reduction [i] would yield (213, 215, large)-net in base 3, but