Best Known (139, 139+32, s)-Nets in Base 8
(139, 139+32, 16384)-Net over F8 — Constructive and digital
Digital (139, 171, 16384)-net over F8, using
- 82 times duplication [i] based on 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
- t-expansion [i] based on digital (136, 169, 16384)-net over F8, using
(139, 139+32, 225510)-Net over F8 — Digital
Digital (139, 171, 225510)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8171, 225510, F8, 32) (dual of [225510, 225339, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8171, 262159, F8, 32) (dual of [262159, 261988, 33]-code), using
- construction XX applied to Ce(32) ⊂ Ce(29) ⊂ Ce(28) [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(8151, 262144, F8, 29) (dual of [262144, 261993, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(81, 14, F8, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(32) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(8171, 262159, F8, 32) (dual of [262159, 261988, 33]-code), using
(139, 139+32, large)-Net in Base 8 — Upper bound on s
There is no (139, 171, large)-net in base 8, because
- 30 times m-reduction [i] would yield (139, 141, large)-net in base 8, but