Best Known (137, 137+32, s)-Nets in Base 8
(137, 137+32, 16384)-Net over F8 — Constructive and digital
Digital (137, 169, 16384)-net over F8, using
- t-expansion [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
(137, 137+32, 196315)-Net over F8 — Digital
Digital (137, 169, 196315)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8169, 196315, F8, 32) (dual of [196315, 196146, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8169, 262150, F8, 32) (dual of [262150, 261981, 33]-code), using
- 1 times truncation [i] based on linear OA(8170, 262151, F8, 33) (dual of [262151, 261981, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [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(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(81, 7, F8, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(8170, 262151, F8, 33) (dual of [262151, 261981, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8169, 262150, F8, 32) (dual of [262150, 261981, 33]-code), using
(137, 137+32, large)-Net in Base 8 — Upper bound on s
There is no (137, 169, large)-net in base 8, because
- 30 times m-reduction [i] would yield (137, 139, large)-net in base 8, but