Best Known (138, 171, s)-Nets in Base 8
(138, 171, 16384)-Net over F8 — Constructive and digital
Digital (138, 171, 16384)-net over F8, using
- 82 times duplication [i] based on digital (136, 169, 16384)-net over F8, using
- net defined by OOA [i] based on linear OOA(8169, 16384, F8, 33, 33) (dual of [(16384, 33), 540503, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(8169, 262145, F8, 33) (dual of [262145, 261976, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(8169, 262145, F8, 33) (dual of [262145, 261976, 34]-code), using
- net defined by OOA [i] based on linear OOA(8169, 16384, F8, 33, 33) (dual of [(16384, 33), 540503, 34]-NRT-code), using
(138, 171, 158969)-Net over F8 — Digital
Digital (138, 171, 158969)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8171, 158969, F8, 33) (dual of [158969, 158798, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8171, 262153, F8, 33) (dual of [262153, 261982, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(8169, 262144, F8, 33) (dual of [262144, 261975, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(8157, 262144, F8, 30) (dual of [262144, 261987, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(8171, 262153, F8, 33) (dual of [262153, 261982, 34]-code), using
(138, 171, large)-Net in Base 8 — Upper bound on s
There is no (138, 171, large)-net in base 8, because
- 31 times m-reduction [i] would yield (138, 140, large)-net in base 8, but