Best Known (61−15, 61, s)-Nets in Base 8
(61−15, 61, 1172)-Net over F8 — Constructive and digital
Digital (46, 61, 1172)-net over F8, using
- net defined by OOA [i] based on linear OOA(861, 1172, F8, 15, 15) (dual of [(1172, 15), 17519, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(861, 8205, F8, 15) (dual of [8205, 8144, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(860, 8204, F8, 15) (dual of [8204, 8144, 16]-code), using
- trace code [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- trace code [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(860, 8204, F8, 15) (dual of [8204, 8144, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(861, 8205, F8, 15) (dual of [8205, 8144, 16]-code), using
(61−15, 61, 8206)-Net over F8 — Digital
Digital (46, 61, 8206)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(861, 8206, F8, 15) (dual of [8206, 8145, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(860, 8204, F8, 15) (dual of [8204, 8144, 16]-code), using
- trace code [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- trace code [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- linear OA(860, 8205, F8, 14) (dual of [8205, 8145, 15]-code), using Gilbert–Varšamov bound and bm = 860 > Vbs−1(k−1) = 11756 957881 522657 339379 051881 811733 219352 199992 293376 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(860, 8204, F8, 15) (dual of [8204, 8144, 16]-code), using
- construction X with Varšamov bound [i] based on
(61−15, 61, large)-Net in Base 8 — Upper bound on s
There is no (46, 61, large)-net in base 8, because
- 13 times m-reduction [i] would yield (46, 48, large)-net in base 8, but