Best Known (81−14, 81, s)-Nets in Base 8
(81−14, 81, 74899)-Net over F8 — Constructive and digital
Digital (67, 81, 74899)-net over F8, using
- 81 times duplication [i] based on digital (66, 80, 74899)-net over F8, using
- net defined by OOA [i] based on linear OOA(880, 74899, F8, 14, 14) (dual of [(74899, 14), 1048506, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(880, 524293, F8, 14) (dual of [524293, 524213, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(880, 524294, F8, 14) (dual of [524294, 524214, 15]-code), using
- trace code [i] based on linear OA(6440, 262147, F64, 14) (dual of [262147, 262107, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(6440, 262147, F64, 14) (dual of [262147, 262107, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(880, 524294, F8, 14) (dual of [524294, 524214, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(880, 524293, F8, 14) (dual of [524293, 524213, 15]-code), using
- net defined by OOA [i] based on linear OOA(880, 74899, F8, 14, 14) (dual of [(74899, 14), 1048506, 15]-NRT-code), using
(81−14, 81, 524296)-Net over F8 — Digital
Digital (67, 81, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(881, 524296, F8, 14) (dual of [524296, 524215, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(880, 524294, F8, 14) (dual of [524294, 524214, 15]-code), using
- trace code [i] based on linear OA(6440, 262147, F64, 14) (dual of [262147, 262107, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(6440, 262147, F64, 14) (dual of [262147, 262107, 15]-code), using
- linear OA(880, 524295, F8, 13) (dual of [524295, 524215, 14]-code), using Gilbert–Varšamov bound and bm = 880 > Vbs−1(k−1) = 12464 788604 206651 475007 771760 498030 434397 827122 368957 146249 193316 220928 [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(880, 524294, F8, 14) (dual of [524294, 524214, 15]-code), using
- construction X with Varšamov bound [i] based on
(81−14, 81, large)-Net in Base 8 — Upper bound on s
There is no (67, 81, large)-net in base 8, because
- 12 times m-reduction [i] would yield (67, 69, large)-net in base 8, but