Best Known (34, 34+6, s)-Nets in Base 8
(34, 34+6, 699118)-Net over F8 — Constructive and digital
Digital (34, 40, 699118)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 65)-net over F8, using
- net defined by OOA [i] based on linear OOA(84, 65, F8, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
- digital (30, 36, 699053)-net over F8, using
- net defined by OOA [i] based on linear OOA(836, 699053, F8, 6, 6) (dual of [(699053, 6), 4194282, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(836, 2097159, F8, 6) (dual of [2097159, 2097123, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(829, 2097152, F8, 5) (dual of [2097152, 2097123, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(836, 2097159, F8, 6) (dual of [2097159, 2097123, 7]-code), using
- net defined by OOA [i] based on linear OOA(836, 699053, F8, 6, 6) (dual of [(699053, 6), 4194282, 7]-NRT-code), using
- digital (1, 4, 65)-net over F8, using
(34, 34+6, 1398106)-Net in Base 8 — Constructive
(34, 40, 1398106)-net in base 8, using
- net defined by OOA [i] based on OOA(840, 1398106, S8, 6, 6), using
- OA 3-folding and stacking [i] based on OA(840, 4194318, S8, 6), using
- discarding factors based on OA(840, 4194320, S8, 6), using
- trace code [i] based on OA(6420, 2097160, S64, 6), using
- discarding parts of the base [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- trace code [i] based on OA(6420, 2097160, S64, 6), using
- discarding factors based on OA(840, 4194320, S8, 6), using
- OA 3-folding and stacking [i] based on OA(840, 4194318, S8, 6), using
(34, 34+6, 6243934)-Net over F8 — Digital
Digital (34, 40, 6243934)-net over F8, using
(34, 34+6, large)-Net in Base 8 — Upper bound on s
There is no (34, 40, large)-net in base 8, because
- 4 times m-reduction [i] would yield (34, 36, large)-net in base 8, but