Best Known (141−32, 141, s)-Nets in Base 8
(141−32, 141, 2048)-Net over F8 — Constructive and digital
Digital (109, 141, 2048)-net over F8, using
- t-expansion [i] based on digital (108, 141, 2048)-net over F8, using
- net defined by OOA [i] based on linear OOA(8141, 2048, F8, 33, 33) (dual of [(2048, 33), 67443, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(8141, 32769, F8, 33) (dual of [32769, 32628, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, 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(8141, 32769, F8, 33) (dual of [32769, 32628, 34]-code), using
- net defined by OOA [i] based on linear OOA(8141, 2048, F8, 33, 33) (dual of [(2048, 33), 67443, 34]-NRT-code), using
(141−32, 141, 28173)-Net over F8 — Digital
Digital (109, 141, 28173)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8141, 28173, F8, 32) (dual of [28173, 28032, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, 32773, F8, 32) (dual of [32773, 32632, 33]-code), using
- 1 times truncation [i] based on linear OA(8142, 32774, F8, 33) (dual of [32774, 32632, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(8141, 32768, F8, 33) (dual of [32768, 32627, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(8136, 32768, F8, 31) (dual of [32768, 32632, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(81, 6, F8, 1) (dual of [6, 5, 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(8142, 32774, F8, 33) (dual of [32774, 32632, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, 32773, F8, 32) (dual of [32773, 32632, 33]-code), using
(141−32, 141, large)-Net in Base 8 — Upper bound on s
There is no (109, 141, large)-net in base 8, because
- 30 times m-reduction [i] would yield (109, 111, large)-net in base 8, but