Best Known (133−25, 133, s)-Nets in Base 8
(133−25, 133, 21847)-Net over F8 — Constructive and digital
Digital (108, 133, 21847)-net over F8, using
- 82 times duplication [i] based on digital (106, 131, 21847)-net over F8, using
- net defined by OOA [i] based on linear OOA(8131, 21847, F8, 25, 25) (dual of [(21847, 25), 546044, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8131, 262165, F8, 25) (dual of [262165, 262034, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8131, 262166, F8, 25) (dual of [262166, 262035, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(8127, 262144, F8, 25) (dual of [262144, 262017, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8109, 262144, F8, 21) (dual of [262144, 262035, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(84, 22, F8, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,8)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(8131, 262166, F8, 25) (dual of [262166, 262035, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8131, 262165, F8, 25) (dual of [262165, 262034, 26]-code), using
- net defined by OOA [i] based on linear OOA(8131, 21847, F8, 25, 25) (dual of [(21847, 25), 546044, 26]-NRT-code), using
(133−25, 133, 205249)-Net over F8 — Digital
Digital (108, 133, 205249)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8133, 205249, F8, 25) (dual of [205249, 205116, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, 262145, F8, 25) (dual of [262145, 262012, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8133, 262145, F8, 25) (dual of [262145, 262012, 26]-code), using
(133−25, 133, large)-Net in Base 8 — Upper bound on s
There is no (108, 133, large)-net in base 8, because
- 23 times m-reduction [i] would yield (108, 110, large)-net in base 8, but