Best Known (32, 32+7, s)-Nets in Base 8
(32, 32+7, 174764)-Net over F8 — Constructive and digital
Digital (32, 39, 174764)-net over F8, using
- 81 times duplication [i] based on digital (31, 38, 174764)-net over F8, using
- net defined by OOA [i] based on linear OOA(838, 174764, F8, 7, 7) (dual of [(174764, 7), 1223310, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(838, 524293, F8, 7) (dual of [524293, 524255, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 524294, F8, 7) (dual of [524294, 524256, 8]-code), using
- trace code [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 524294, F8, 7) (dual of [524294, 524256, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(838, 524293, F8, 7) (dual of [524293, 524255, 8]-code), using
- net defined by OOA [i] based on linear OOA(838, 174764, F8, 7, 7) (dual of [(174764, 7), 1223310, 8]-NRT-code), using
(32, 32+7, 524296)-Net over F8 — Digital
Digital (32, 39, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(839, 524296, F8, 7) (dual of [524296, 524257, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(838, 524294, F8, 7) (dual of [524294, 524256, 8]-code), using
- trace code [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- linear OA(838, 524295, F8, 6) (dual of [524295, 524257, 7]-code), using Gilbert–Varšamov bound and bm = 838 > Vbs−1(k−1) = 5 548501 409905 442638 665188 914287 [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(838, 524294, F8, 7) (dual of [524294, 524256, 8]-code), using
- construction X with Varšamov bound [i] based on
(32, 32+7, large)-Net in Base 8 — Upper bound on s
There is no (32, 39, large)-net in base 8, because
- 5 times m-reduction [i] would yield (32, 34, large)-net in base 8, but