Best Known (33, 33+7, s)-Nets in Base 8
(33, 33+7, 174768)-Net over F8 — Constructive and digital
Digital (33, 40, 174768)-net over F8, using
- net defined by OOA [i] based on linear OOA(840, 174768, F8, 7, 7) (dual of [(174768, 7), 1223336, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(840, 524305, F8, 7) (dual of [524305, 524265, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(840, 524306, F8, 7) (dual of [524306, 524266, 8]-code), using
- trace code [i] based on linear OA(6420, 262153, F64, 7) (dual of [262153, 262133, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(6419, 262145, F64, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(6413, 262145, F64, 5) (dual of [262145, 262132, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(647, 8, F64, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
- dual of repetition code with length 8 [i]
- linear OA(641, 8, F64, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- Reed–Solomon code RS(63,64) [i]
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- trace code [i] based on linear OA(6420, 262153, F64, 7) (dual of [262153, 262133, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(840, 524306, F8, 7) (dual of [524306, 524266, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(840, 524305, F8, 7) (dual of [524305, 524265, 8]-code), using
(33, 33+7, 524306)-Net over F8 — Digital
Digital (33, 40, 524306)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(840, 524306, F8, 7) (dual of [524306, 524266, 8]-code), using
- trace code [i] based on linear OA(6420, 262153, F64, 7) (dual of [262153, 262133, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(6419, 262145, F64, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(6413, 262145, F64, 5) (dual of [262145, 262132, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(647, 8, F64, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
- dual of repetition code with length 8 [i]
- linear OA(641, 8, F64, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- Reed–Solomon code RS(63,64) [i]
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- trace code [i] based on linear OA(6420, 262153, F64, 7) (dual of [262153, 262133, 8]-code), using
(33, 33+7, large)-Net in Base 8 — Upper bound on s
There is no (33, 40, large)-net in base 8, because
- 5 times m-reduction [i] would yield (33, 35, large)-net in base 8, but